114 Mr WALLACE, ON GEOMETRICAL THEOREMS, AND FORMULA, 



where it must be observed that the angles a, fi, y, must be so reckoned, 

 that theu- sum is four right angles. 



By a known Trigonometrical formula, 



a;-- 2iry cos 7 +y- = c- (1), 



af-2xz cos(i + s' = b- (2), 



y-—2i/x cos a +a- = a? (3). 



The condition that the triangle ^BC is made up of the three tri- 

 angles ADB, ADC, BDC (fig. 5), or else of the excess of two of 

 them above the third (fig. 6), is expressed analytically by this other 

 equation, 



xy sin y + xx sin ji + y% sin a=^bc sin A (4). 



These hold true, whatever be the position of the point D, observing 

 always that the angles a, /3, 7 must satisfy the condition of making up 

 four right angles. 



By adding the first and second equations, and subtracting the third, 

 we obtain, 



^x'^ — Ixy cos y — 'ixx cos /3 + %yz cos a = h'' -^^ & — a". 



But b' + c- — a" = 2bc cos A ; therefore, 



• x^ — xy cos y — xz cos li + y~ cos a — be cos A (5). 



Let equation (5) be multiplied by sin a, and equation (4) by cos a : the 

 results are 



x' sin a — xy sin « cos 7 — xs sin a cos (i + yz sin a cos a = bc sin a cos A, 



xy cos a sin 7 + xz cos a sin fi + yx sin o cos a = be cos a sin A. 



By subtracting the second of these equations from the first, and ob- 

 serving that 



