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These facts suggest two ways to determine the equation of the median median line They are as follows: 1 First, determine the slope of the line joining P1 and P3 This is the slope of the median median line Second, determine the point (x, y) Finally, nd the median median line using the slope and a point on the line 2 Determine the slope of the line joining P1 and P3 Then determine the equations of two of the medians and solve them simultaneously to determine the point of intersection The median median line can be found using the slope and a point on the line To these two methods, suggested by the facts above, we add a third method 3 Determine the intercept of the line joining P1 and P3 and the intercept of the line through P2 with the slope of the line through P1 and P3 The intercept of the median median line is the average of twice the rst intercept plus the second intercept (and its slope is the slope of the line joining P1 and P3 ) Method 1 is by far the easiest of the three methods although all are valid Methods 2 and 3 are probably useful only if one wants to practice nding equations of lines and doing some algebra! Method 2 is simply doing the proof above with actual data We will not prove that method 3 is valid but the proof is fairly easy

(4) BaseAdd res s : RegionS i z e : State :

Protect : PAGE_R EADWRITE + PAG E_GUARD Type : aaa2aaaa MEM_PR IVATE (5) B a s eAddres s : RegionS i z e : State : P rotect : Type :

This chapter showed how to create journal entries for any payroll situation that is likely to arise, and noted the differences between journal entries derived from inhouse versus outsourced payroll systems A key issue noted several times is the prevention of inconsistencies in the reporting of information through journal entries this can be a major problem, especially since payroll costs can comprise the bulk of all company expenses Consequently, pay close attention to the prevention of payroll transactional errors

B a s eAdd re s s : aaaaaaaaaa 182aaa RegionS i z e : aaaaaaaaaaaa7aaa State : aaaalaaa MEM_COMMIT P rotec t : aaaaala4 PAG E_R EADWRITE + PAG E_GUARD Type : aaa2aaaa MEM_PRIVATE

It turns out that if x2 = x, then the least squares line and the median median line are identical To show this, consider the diagram in Figure 139 where we have taken the base of the triangle along the x-axis and the vertex of the triangle at (x1 /2, y2 ) since 0 + x1 + 3 x1 2 = x1 2

aaaaaaaaaa17eaaa B a s eAdd res s : aaaaaaaaaaaa2aaa RegionS i z e : aaaalaaa MEM_COMMIT State : aaaaaaa4 PAG E_R EADW R I T E P rotec t : aaa2aaaa MEM_PR IVAT E Type :

aaaaaaaaaa 179aaa aaaaaaaaaaaa7aaa aaaalaaa MEM_COMMIT aaaaaaa4 PAG E_R EADWR ITE aaa2aaaa MEM_PRIVAT E

1 This journal entry was derived with permission from Appendix B of Bragg, Accounting Reference Desktop (Hoboken, NJ: John Wiley and Sons, 2002) 2 Ibid 3 Ibid 4 Ibid

.

(x1/2,y2)

In native code, there are three distinct regions (2, 4, and 5), and in man aged code there are five Let's inspect each in detail Because the stack grows downward in the address space, we'll discuss them in the reverse order: 5 The actively used portion of the stack It is fully committed, backed by the pagefile, and several pages are probably (but not necessarily) resident in RAM Notice that the Ba seAd d r e s s is equal to the thread's current St a c k L imit, and that B a s eAd d r e s s + Regio n S i z e equals St a c k B a s e This i s a basic invariant The thread i s actively reading from and writing to its stack memory only within this region, and the E S P register is likely pointing inside of it unless stack growth is imminent 4 The guard region of the stack Notice that its protection attributes include PAG E_GUARD, and that it too is committed When the stack grows into the guard region, the current pages inside the guard will become part of region 5, and the next pages further down in the stack will become the new guard region A few things are worth noting Notice that the guard page is a single page in the native case, but its Regio n S i z e is ex7eee (28KB) in managed That's because the CLR always uses the SetTh r e a d St a c kG u a r a nt e e for managed threads on OSs that support it It does this in order to make responding to stack overflow and shutting down the CLR cleanly possible 3 This is the last page of the used portion of the stack and will never truly be committed It's often referred to as the "hard guard page".