740 MR EDWARD SANG ON THE THEORY OF COMMENSURABLES. 



But 



wherefore 

 and 



DG=DH+HE+EF ; AG=AF-EH, 



DG = ba + aa + aj3 ; AG=b^—aa, 



P. Q=(&a + aa + a/5)2 +(&a + aa + a/3) {b3 — aa)-\-{h(B — aaf. 



Again ; make the angle EAD' equal to EAD, AD' equal to AD ; join ED', draw 

 G'D'I' parallel to GD, EI' parallel to FG' and D'H' to EH ; then it is easy to show 

 that D'l'H' is equilateral, and that ED'H' is equal to EDH. 



Now 



D'G'=EF-D'H' , AG'=AF + EH + HD, 

 wherefore 



and 



D'G'=a/3 — &a ; AG' = 6/3 + aa + ia, 



P . (^={a^ — haf + {a^—ba) {b(3 ■*- aa + ba) + (b^ + aa + baf , 



thus giving a second decomposition of PQ. 



By interchanging a for b and a for /5, in the above expressions, we obtain 

 other two, viz., — 



P.Q=(Z>a-a/3/ + (6a-o/3) (b(3 + al3 + aa) + (b/S-^ a[3 + aay 

 T .q={aa-b(3f + (aa-bl3) (ba + a(3 + bl3) + (ba + al3+b(3y . 



and at first we might suppose that there are four decompositions : but if we take 

 notice that if ha—a^ be positive in the one it is negative in the other, so that two 

 of the four must belong to the form A-— AB + B'. 



If we regard the trigon ABC as analogous to the right-angled trigon of the 

 previous part, we may, taking AB as the unit or radius, consider BC as coming 

 in place of the sine, AC in place of the cosine ; and, for the moment, we may call 

 the former the opposite, the latter the adjacent of the angle: that is v/e may 

 define 



-T-p as the opposite of CAB 



AC 



-T-j3 as the adjacent of CAB ; 



and then we form four theorems for the functions of the sum and difference of 

 two angles, analogous to the fundamental theorems of trigonometry. These are, 

 putting and ^ for the angles BAC, /3A7, 



opp. (0+^}=opp. (p. adj. ^ + adj. <p. opp. ^ + opp. <p opp. 6 

 adj. (0 + 0) = adj. (p. adj. — opp. (p. opp. 6 



opp. (0 — 0) = opp. (p. adj. 0— adj. (p. opp. 6 



adj. ((/) — ^) = adj. (p. adj. ^ + opp. (p. opp. ^ + adj. (p. opp. 6. 



in which it is to be remarked that we cannot pass from the function of <p + d, to 



