# NC Political Firebombing Not a Recent First

It’s scary, all right:

Gov. Pat McCrory Sunday called the weekend firebombing of a North Carolina Republican headquarters “an attack on our democracy,” while one GOP official called it an act of “political terrorism.”

In a tweet, Republican presidential candidate Donald Trump blamed “Animals representing Hillary Clinton and Dems in North Carolina.”

Hillsborough police said somebody threw a bottle of flammable liquid through the window of Orange County’s GOP headquarters, setting supplies and furniture ablaze.

But not without precedent:

Just 5 days before the election, at 3a.m. on October 30th, all of the front windows of the Cindy Sheehan for Congress campaign offices were shattered. Although staffers had been in the office less than an hour earlier, no one was in the building at the time of the incident. No one was hurt and there were no witnesses. Cindy Sheehan is a candidate for Congress in California’s 8th Congressional District race against incumbent Nancy Pelosi (D-CA).

“It seems to have been a calculated intimidation tactic,” said Tiffany Burns, the Cindy for Congress campaign manager. “One of our computers was stolen, but no other property was taken from our offices and no surrounding buildings were targeted. Clearly they wanted to both frighten us and to gather information.” Total damage to the campaign office is currently estimated at more than \$5,000.

As I said, thugocracy isn’t fun…not for anyone.

# A Little Lesson in Subsidarity

One of pot shots that Hillary Clinton and her operatives made at conservative Catholics is that they used terms like “subsidarity” that no one understood. Since they may be right about that, I think an illustration is in order.

Many of you know that I teach Civil Engineering. Six years ago, my department head (who is from Kenya) and his first assistant (who is from the Cameroon) sat me down and asked me to obtain my PhD so I could teach more courses. I agreed and six years later, as W.H Auden said about Tolkien, at the end of the quest, victory.

In the course of the conversation, my department head brought up the subject of why potholes don’t get fixed in Africa the way they do here. (I know we have issues here.) His explanation was this: here, the local authorities (city, county, state) maintain the roads and, since they’re closer to the problem, they have greater incentive to fix it. Back home, decisions are made in the capital, and since they’re far away from the roads, they don’t have a pressing interest, and the potholes remain. That’s probably the best illustration of the concept of subsidarity—which seeks to push decision-making down to the lowest level—that I’ve heard.

Roman Catholicism—especially in its Ultramontane form, which has been the norm since the Restoration—is not the most suitable vehicle to promote the idea of subsidarity. It’s a good theological concept, but the structure of the church works against it.

As far as Hillary Clinton is concerned, truth be told, her problem with subsidarity isn’t that she doesn’t understand it. Her problem is that she doesn’t like it. Her idea—one that has been obvious since Arkansas’ educational “reforms” in the 1980’s—is that power and decision-making be concentrated at the top. People who support subsidarity are political enemies, which is a big reason she wants a “Catholic Spring.”

As far as how two Africans got a Palm Beacher like me to pursue a PhD, it’s another sign that, in engineering, we really do have change we can believe in.

# Americans Aren’t the Only Ones to Choose Between “Bad” and “Really Bad”

All of the blubbering and whining going on about how Americans (and especially Christians) are somehow criminal to vote for a certain candidate needs to be put in this perspective, from a 2013 post:

A few months ago, when the Iranians elected themselves a new president, I asked an Iranian friend what he thought of the election.  His response was simple: the Iranian people had a choice between bad and really bad in the election and chose only bad.

Let me ask this: have any of our self-appointed moral compasses in the Christian community ever gone to Iran to advise our brethren there not to vote for a certain candidate because he favours continuing forcing  women to wear the hijab, or hanging homosexuals from construction equipment, or supporting the nuclear weapons program?  Of course not; none of these people would even try to get past Iran’s security system to make this appeal.  And most of them are clueless about societies where these kinds of decisions are a part of daily survival.

The difference between Iran and the United States when it comes to elections is that nominations in Iran are managed by the ruling Islamic system, where here nominations are managed by the stupid nominating system we have.  And the DNC, by tilting the table towards Hillary Clinton, showed that the ayatollahs in Tehran have met their match when it comes to management.  Be that as it may, we have the absurd situation where both major parties have nominated candidates that most Americans don’t like, and that’s a stupid nominating system.

Nine years ago I made it clear that I cannot vote for Hillary Clinton.  That hasn’t changed.  If those who cannot understand the limitations of electoral process–ours and others–don’t understand that, there’s not much left to say.

# Our Elites’ Snotty Attitudes, Then and Now

Wikileaks’ revelations that Hillary Clinton and her operatives take a dim view of social conservatives–following her characterization of large portions of the population as “deplorables”–has ignited a great deal of anger.  As someone who started out life growing up with the elites, I think some perspective is in order.

Let me start by putting up something that’s been on this website since 2004.  It comes from my Around the Island post about Palm Beach, and it goes like this:

Below: “There’s a hole in my bucket…” Fourth graders at Palm Beach Day School perform a satire on “hillbillies” called “Appalachian Legend” during Stunt Night 1969. Attitudes from the “coasts” about “flyover country” in the U.S. have been deep seated for a long time; stage productions like this only reinforced that. It’s fair to say that, if the “Religious Right” had fully grasped the contempt they were held in when the movement first got going in the late 1970’s they would not have started the Moral Majority: they would have started a revolution.

Haughty attitudes of our elites towards the rest of the population aren’t new; they’re as old as class differences.  So why didn’t the revolution I thought would be a logical outcome (then, at least) not happen?  There are several reasons:

1. Our elites had better taste and manners then; they knew better than to rub the rest of the population’s face in their perceived superiority.
2. We lacked the instant means of communicating contempt we have now.
3. Most of the “moral majority” didn’t see the difference between their values and those at the top as class based.   That was simply false; the top of our society had been lost to the fervent Evangelicalism for a long time, being steeped in either Main Line Christianity or Judaism.
4. Some actually did, but didn’t care; they felt that those at the top would go to hell for their lack of belief and they would not.  That kind of “remnant” mentality was very deep in Evangelical Christianity, especially in the South.  One result of the political activity of the last forty years or so is the erosion of that mentality.
5. Others sensed it, but were too ashamed to admit it, because it would imply those who opposed them were better than they were.  They were and are the aspirational types; much of the impetus for political involvement has come from these people.
6. Income inequality has increased since that photo was made; the gap between the elites and the Appalachians has grown significantly.

• I think it strange that the standard-bearer of those who seek a revolution is a billionaire; it’s one of those bizarre American things.  But it’s the aspirational way: those who idolize Trump project their own aspirations into his own success, which is very common in our society.
• On the other hand, aspirational people are a threat to the existing power holders, which is why Hillary Clinton and her operatives feel about them the way they do.  Elites, then and now, prefer corporatism.  And that’s ironic too for a bunch whose ideological roots are in 1960’s radicalism.
• As far as her attitudes towards social conservatives is concerned, what we’re headed for under her idea is a “two-tier” religious structure where certain churches and religious organizations are “acceptable” and certain ones are not, with legal disabilities following.  That was the case in Nazi Germany with the “Confessing Church,” in the Soviet Union, and is the case in China, although the Three-Self Church is showing many signs of life.  Her idea that Roman Catholicism is an “élite” religion (as opposed to Evangelicalism) has a strange feel to it.  Going from Episcopal to Catholic was a drop in social level in the 1970’s, but the Main Line churches have lost most of their relevance even at the top.
• Trump’s crudity is unsurprising, especially for someone raised in South Florida as I am.  What we have to choose from is one candidate whose forced sexualization agenda is one of personal depravity and the other whose forced sexualization agenda is a matter of public policy.
• Personally I’ve always gravitated to the “remnant” mentality.  I was raised listening to the encounter with the rich young ruler and the parable of the rich man and Lazarus; somehow anything else misses the point.  The most active alternative along these lines is the “Benedict Option” advocated by Rod Dreher.  Maintaining that in a totalitarian society–even one with periodic elections–won’t be easy.

# Buoyancy and Stability: An Introduction

Ever since people set out to sea in ships, the issues of buoyancy and stability have been of importance. In spite of this, the treatment it receives in textbooks is often lacking. Following is an overview of the subject; basic understanding of the principles is essential in performing the experiment and interpreting the results.

## Buoyancy

Buoyancy is ultimately what makes things float, such as the buoy in Figure 1. This is true whether the material the boat is made of is lighter than water (like the balsa wood rafts Thor Heyerdahl and his crew crossed the Pacific with in 1947) or heavier than water. The latter would include objects from the buoy shown to the ships of the U.S. Navy.

The basic concept is very simple: for anything placed in a fluid medium, the upward force the medium exerts on the body is equal to the weight of the fluid the body displaces. This is not only true of bodies placed in water; it is also true of those in air. The difference is that, for those in air, the weight of the air displaced is usually not enough to “float” the aircraft. A notable exception are dirigibles such as the “Goodyear blimp,” which is filled with helium, a gas lighter than air. Another lighter-than-air gas used is hydrogen. This is very combustible, as everyone was reminded of when the Hindenburg caught fire in New Jersey in 1938.

Most buoyancy applications are marine ones, and it is those we will concentrate on in this experiment. We will also concentrate on rectangular forms and flat-bottomed vessels, which simplifies the math somewhat. However, these principles can be extended to just about any floating craft.

Using a flat-bottomed craft also makes it easier to understand why displacing a fluid works. Consider first the following: how the force of the fluid on the flat hull of a craft varies with depth1:

For a fluid at rest, the hydrostatic pressure increases linearly with depth, thus

(1)

where p is the hydrostatic pressure, γ is the unit weight of the water, and D is the depth from the water’s surface to the bottommost point of the vessel, usually called the draught. This distance from the water line to the top of the rectangle (the gunwale) is called the freeboard; the results of inadequate freeboard can be seen in Figure 3.

In any case, for a vessel of beam (width) W and a length L the volume it displaces is given by the equation

(2)

Combining and rearranging these two equations,

(3)

For the boat to float, it has to be in static equilibrium, and so the downward force of the weight of the boat Wboat must equal the upward force Fbuoyant. Therefore,

(4)

So we’ve established a relationship between the weight of the boat and the volume of water it displaces. The “far right” hand side only applies to boats with a flat bottom and straight sides.

What this means is that there are three ways we can weigh an existing boat:

1. We can simply weigh it on a scale. For small boats this isn’t too difficult; larger ones can be tricky.  We can then estimate how far it will sink into the water.

2. We can measure the freeboard, then obtain D and, knowing L, W and the unit weight of water, we can compute the weight of the boat. This works easily for rectangular boats; for real boats, you have to determine the relationship between the actual waterline and the displacement, then see where the actual waterline ends up.

3. We can use an overflow method, which is okay for small experiments (like Archimedes used) but not so hot on a larger scale.  But this illustrates our concept.

Procedure for determining volume of water displacement2:

## Stability

Buoyancy is a fairly straightforward concept, although it may be a little hard to grasp up front. Stability—the ability of the ship to resist overturning—is a little more difficult, although it’s obviously important, as the following diagram of a ship with waves coming at the beam shows3.

Let’s define (or recall) a couple of terms.

Centre of Gravity: this is easy, mathematically this is the centroid of the mass or weight of the ship. An illustration of this is below.

Centre of Buoyancy: this is a little trickier, this is the centroid of the cross-sectional area of the ship under the water line, as shown below.

As you can see, for a box-shaped vessel which is not listing (i.e., leaning at an angle) or has no squat (i.e., not angled along the length of the boat) the centre of buoyancy is located halfway down the draught of the vessel, halfway across the beam, and dead amidships.

The centre of gravity and the centre of buoyancy are not necessarily at the same place; in fact, they are usually different. That difference determines both the stability of the ship and, literally, how it rolls.

We know that motor vehicles with high centres of gravity (such as off-road vehicles) are more prone to turn over in use than those with lower centres of gravity. Ships are the same; we need to have a way to decide how stable a ship is and whether there is a point that a ship becomes unconditionally stable or unconditionally unstable.

As long as a ship is upright, and both the centre of gravity and the centre of buoyancy are in the centre of the ship in all respects, it is theoretically possible for a ship never to turn over. As a practical matter this is impossible; even very large ships like cruise ships, which use their size to resist roll in most wave situations, are going to roll some. Below is a diagram which shows the centre of gravity and the centre of buoyancy for a ship which is upright and which is inclined 14º.

We need to look at this carefully and note the following:

• The point G is the centre of gravity of the ship.

• The point B or B’ is the centre of buoyancy of the ship. In the course of inclination the centre of buoyancy will change because the shape of the cross-section under the waterline changes; this is fairly simple to calculate for rectangular ships and more complicated for curved hull shapes.

• The point M is the metastatic point of the ship. The distance GM is called the metastatic height of the ship.

• If point G is below point B or B’, the ship is unconditionally stable; it will not turn over unless G and B’ is changed by taking on water, shifting cargo in the ship, etc.

• If point G is below point M, the ship is conditionally stable, and if point G is above point M, the ship is unconditionally unstable.

The reason for this last point is simple: the ship above is rolling in a clockwise direction. The resisting moment of the buoyancy, calculated by (GZ)(Wbuoyant) is counter-clockwise, as the buoyant force is upward. This is true as long as G is below M. If G moves upward above M, then the now driving moment (GZ)(Wbuoyant) turns clockwise, the same direction as the rolling of the ship, and the ship will generally turn over4.

Thus the location of M, abstract as it may seem, becomes a critical part of the design of a ship. But how is it done? There are two methods we will discuss here of determining the metastatic height of a ship.

## Determining Metastatic Height

### Theoretical Method

This method uses the following formula to determine the location of the metacentre:

(5)

For a rectangular vessel, the moment of inertia is the same as we used in mechanics of materials, i.e., LW3/12, and is applied as follows:

The displacement volume was given earlier. We then compute the distance between the metacentre M and the centre of buoyancy B as follows:

Note carefully that this is NOT the metacentric height GM; it is then necessary to subtract the distance from the centre of buoyancy to the centre of gravity from this result to obtain GM. This is done as follows:

It’s worth noting here that the location of point M is independent of the centre of gravity and dependent upon the geometry of the ship and its volume under the water line (or total weight.)

### Timing the Roll

This method is sort of an “old salt’s” rule of thumb method. First, let’s define the roll time. The roll time is the time it takes for a ship to start from rest at an angle of roll (port or starboard,) roll to the opposite side, and return to the original orientation. This can be approximated by the equation5

(6)

where tr = roll time of ship, seconds
GM = metastatic height of ship, meters or feet
W = beam of ship, meters or feet
C = constant based on units of GM and B
= 0.44 for units of feet
= 0.80 for units of meters

Solving for metastatic height,

(7)

This is significant for another reason: another rule of thumb used in yacht design is that the roll time is seconds should be between 1 and 1.1 times the beam W of the boat in meters6. Yachts with shorter roll times tend to “check” or quickly come back to centre when in rough seas; this can be a hard experience for passengers and make a complete mess of stowed cargo. Yachts with longer roll times will come over like they’re about to capsize, and then slowly roll back around. The usual result of this is seasickness and a miserable ride.

1Walton, T. (1899) Know Your Own Ship. Charles Griffin and Company, London, England. Much of the material that follows on buoyancy and stability comes from this work.

2Keep in mind that the unit weight of sea water is greater than fresh. Why is this so?

3Seamanship pointer: if you’re in a boat and are facing high waves, wake, etc., best way to take them is to point the blow into the direction the waves are coming from, not take the waves on the beam.

4Whether it actually does gets into issues of freeboard, rate of roll, etc., which are beyond this presentation or experiment.

5Nudelman, N. (1992) Yacht Design Course. Lesson 6: Stability. Westlawn Institute of Marine Technology, Stamford, CT.

6Gerr, D. (1992) The Nature of Boats. International Marine Publishing, Camden, ME. Much of the material on this method of determining GM is taken from this source.

# When Catholic Academia Bails on Philosophy, We’re All in Trouble

A similar crisis has shaken the philosophical estate within the church. Before 1970 philosophy enjoyed an enviable prominence in the curriculum of Catholic colleges. This Neo-Scholastic philosophy was certainly structured around the perennial questions—Does God exist? What is virtue?—but it was an odd, manual Thomism in which students never actually read Aquinas. A smug catechetical certitude seemed to lurk behind the paint-by-numbers proofs and the gleeful one-paragraph refutations of modern “adversaries.”

That world has disappeared; its chastened replacement in the Catholic academy bears the stamp of marginality: minimal curricular presence, hyper-specialization, incoherence among the squabbling philosophical factions.

This cultural recession of philosophy has encouraged some Catholics to abandon philosophy as a central component of the church’s discourse. The issue has become especially neuralgic in the dispute over the formation of clergy. But the project of a nonphilosophical Catholicism is fraught with peril.

In some ways, the greatest blessing God bestowed on me in my Christian formation was to dodge education at one of these institutions, which enabled me to take in Aquinas and the like on the side.  Not only did it avoid the problems there, but inculcation in philosophically structured Christianity has helped me to avoid some of the sillier–and more dangerous–trends in Charismatic and Pentecostal thought.

Catholic theology in particular is pretty much toast outside of a philosophical framework.  And that throws away one of the major advantages that Catholicism has.  That’s a pity, the rest of us need the discipline.

The blunt truth is that Evangelical theology is an oxymoron precisely because it rejects any philosophical framework.  The Bible, however, was written in the flow of human history and experience, where we live, and was intended to address that experience directly.  Sooner or later, however, the question of “why?” will come up, and without a philosophical framework that question is unanswerable.

Today Pentecostal and Charismatic theology is at a crossroads because it inherited Evangelical theology’s basic thought structure without its limiting assumptions.  The result is that some in the academy (and elsewhere) are about to take the leap outside of Christianity without knowing it, and we all know where that ends.

There are problems with philosophy too; I tackled that issue some time back here.  Some of the problems we have now could be avoided by jettisoning much of modern philosophy altogether.  But throwing out the baby with the bath water isn’t the answer, for Catholics or anyone else.

# Getting Closer on the St. Andrew’s Sex Scandal?

Bishop Audrey Scanlan of the Episcopal Diocese of Central Pennsylvania yesterday removed the Rev. Howard White from the priesthood.

White, 75, was among several adults who sexually abused students at St. George’s School in Middleton, Rhode Island in the 1970s and 80s, according to a report released recently by independent investigators on behalf of the school.

In the wake of media reports about sexual abuse at St. George’s, several people from other Episcopal dioceses in which White had worked said that he sexually abused them when they were young.

It’s interesting to note that, according to this, “(Headmaster) Zane terminated White in 1974 after he said White admitted ‘sexually abusing a sophomore boy and attempting to sexually abuse at least two and likely three others.'”  The wheels of justice turn very slowly in this “enlightened” Episcopal Church, since someone was put on notice 42 years ago.  It puts Episcopal Bishop Porter Taylor’s statement that White “…had been identified by former students of St. George’s School in Rhode Island as having engaged in sexual misconduct in the early 1970s while he served on the staff at that school” in a different light.  He should have used the plaintiff attorney’s favourite phrase “by his own admission,” but he didn’t.

So what does this have to do with the scandals at the other end of the East Coast?  As I noted in my last piece on the subject, the link is the Rev. George Andrews II, who went from St. George’s to St. Andrew’s as Headmaster in the late 1980’s.  Will he and other Episcopal reverends be implicated?  We will see.

And as far as that “other” sex scandal we have in the Presidential election, I’ll stand by my previous position that, not so far in the future, such things won’t be scandalous any more, not in our society.  As was the case with the Roman Catholic Church, the left will milk the scandal cow as long as it lives before they butcher it, and Evangelical leadership is just plain clueless.

# Mohr’s Circle Analysis Using Linear Algebra and Numerical Methods

## Abstract

Mohr’s Circle-or more generally the stress equilibrium in solids-is a well known method to analyse the stress state of a two- or three-dimensional solid. Most engineers are exposed to its derivation using ordinary algebra, especially as it relates to the determination of principal stresses and invariants for comparison with failure criteria. In this presentation, an approach using linear algebra will be set forth, which is interesting not only from the standpoint of the stress state but from linear algebra considerations, as the problem makes an excellent illustration of linear algebra concepts from a real geometric system. A numerical implementation of the solution using Danilevsky’s Method will be detailed.

## 1. Introduction

The analysis of the stress state of a solid at a given point is a basic part of mechanics of materials.Although such analysis is generally associated with the theory of elasticity, in fact it is based on static equilibrium, and is also valid for the plastic region as well. In this way it is used in non-linear finite element analysis, among other applications. The usual objective of such an analysis is to determine the principal stresses at a point, which in turn can be compared to failure criteria to determine local deformation.In this approach the governing equations will be cast in a linear algebra form and the problem solved in this way, as opposed to other types of solutions given in various textbooks. Doing it in this way can have three results:

1. It allows the abstract concepts of linear algebra to be illustrated well in a physical problem.
2. It allows the physics of the determination of principal stresses to be seen in a different way with the mathematics.
3. It opens up the problem to numerical solution, as opposed to the complicated closed-form solutions usually encountered, of the invariants, principal stresses or direction cosines.

## 2. Two-Dimensional Analysis

### 2.1. Eigenvalues, Eigenvectors and Principal Stresses

The simplest way to illustrate this is to use two-dimensional analysis. Even with this simplest case, the algebra can become very difficult very quickly, and the concepts themselves obscured. Consider first the stress state shown in Figure 1, with the notation which will be used in the rest of the article.

Figure 1: Stress State Coordinates (modified from Verruijt and van Bars [8])

The theory behind this figure is described in many mechanics of materials textbooks; for this case the presentation of Jaeger and Cook [4] was used. The element is in static equilibrium along both axes. In order for the summation of moments to be zero,

$\tau_{xy}=\tau_{yx}$ (1)

The angles are done a little differently than usual; this is to allow an easier transition when three-dimensional analysis is considered. The direction cosines based on these angles are as follows:

$l=cos\alpha\$ (2)

$m=cos\beta\$ (3)

Now consider the components $p_{x},\,p_{y}$ of the stress vector p on their respective axes. Putting these into matrix form, they are computed as follows:

\left[\begin{array}{cc} \sigma_{{x}} & \tau_{{\it xy}}\\ \noalign{\medskip}\tau_{{\it xy}} & \sigma_{{y}} \end{array}\right]\left[\begin{array}{c} l\\ \noalign{\medskip}m \end{array}\right]=\left[\begin{array}{c} p_{{x}}\\ \noalign{\medskip}p_{{y}} \end{array}\right] (4)

which is a classic Ax = b type of problem. At this point there are some things about the matrix in Equation 4 that need to be noted (DeRusso et al. [2]):

1. It is square.
2. It is real and symmetric. Because of these two properties:
1. The eigenvalues (and thus the principal stresses, as will be shown) are real. Since for two-dimensional space the equations for the principal stresses are quadratic, this is not a “given” from pure algebra alone.
2. The eigenvectors form an orthogonal set, which is important in the diagonalization process.
3. The sum of the diagonal entries of the matrix, referred to as the trace, is equal to the sum of the values of the eigenvalues. As will be seen, this means that, as we rotate the coordinate axes, the trace remains invariant.

At this point there are two related questions that need to be asked. The first is whether static equilibrium will hold if the coordinate axes are rotated. The obvious answer is “yes,” otherwise there would be no real static equilibrium. The stresses will obviously change in the process of rotation. These values can be found using a rotation matrix and multiplying the original matrix as follows (Strang [7]):

\left[\begin{array}{cc} cos\alpha & -sin\alpha\\ sin\alpha & cos\alpha \end{array}\right]\left[\begin{array}{cc} \sigma_{{x}} & \tau_{{\it xy}}\\ \noalign{\medskip}\tau_{{\it xy}} & \sigma_{{y}} \end{array}\right]=\left[\begin{array}{cc} \sigma'_{{x}} & \tau'_{{\it xy}}\\ \noalign{\medskip}\tau'_{{\it xy}} & \sigma'_{{y}} \end{array}\right] (5)

The rotation matrix is normally associated with Wallace Givens, who taught at the University of Tennessee at Knoxville. The primed values represent the stresses in the rotated coordinate system. We can rewrite the rotation matrix as follows, to correspond with the notation given above:^

\left[\begin{array}{cc} l & -m\\ m & l \end{array}\right]\left[\begin{array}{cc} \sigma_{{x}} & \tau_{{\it xy}}\\ \noalign{\medskip}\tau_{{\it xy}} & \sigma_{{y}} \end{array}\right]=\left[\begin{array}{cc} \sigma'_{{x}} & \tau'_{{\it xy}}\\ \noalign{\medskip}\tau'_{{\it xy}} & \sigma'_{{y}} \end{array}\right]\ (6)

### 2.1 Eigenvalues, Eigenvectors and Principal Stresses

The second question is this: is there an angle (or set of direction cosines) where the shear stresses would go away, leaving only normal stresses? Because of the properties of the matrix, the answer to this is also “yes,” and involves the process of diagonalizing the matrix. The diagonalized matrix (the matrix with only non-zero values along the diagonal) can be found if the eigenvalues of the matrix can be found, i.e., if the following equation can be solved for $\lambda$ :

\left[\begin{array}{cc} \sigma_{{x}}-\lambda & \tau_{{\it xy}}\\ \noalign{\medskip}\tau_{{\it xy}} & \sigma_{{y}}-\lambda \end{array}\right]=0\ (7)

To accomplish this, we take the determinant of the left hand side of Equation 7, namely