438 



Prof. Sylvester on a Memoria Teclmica. 



But even without this check the false equations may be put 

 to the question and made severally to disclose their character 

 as such by applying any one of them to the limiting* case of a 

 triangle on a sphere continuing of finite radius, but in which the 

 angles become respectively 180°, 0, 0, and consequently the side 

 opposite the first equal or capable of being equal to the sum of 

 the other two. Thus writing in the first and third of the last 

 written formulae C = 180°, B = 0, A = 0, we ought to be able to 

 derive c—a-\-b, but find instead a = b + c in the first, and 

 a=b + c in the third. And similarly in the second and fourth, 

 writing A =180°, B = 0, C = 0, we ought to be able to derive 

 a= b -f c, but find instead a = — b + c in the second, and 

 a= —b-\-c, or a + b + c=360° in the fourth. We might easily 

 deduce other detective criteria from the reciprocal limiting case 

 of a spherical triangle in which one side is zero and the two 

 others each 180°, in which case the angle opposite the first aug- 

 mented by 180° will equal the sum of the other two. Further- 

 more, using accents, as before, to denote polar reciprocation, the 

 false system takes the form 



P-P=0, Q + B/=0, B + Q' = 0, S-S'=0, 



in lieu of the true form, 



P + p = 0, Q-B/=0, B-Q' = 0, S + S'=0. 



A direct geometrical proof of these potent formulae appears to 

 be a desideratum, \ 



K House, Woolwich Common, 

 November 9, 1866. 



aid of the scheme below written, 



left, right. 



+ 



- 



- 



+ 



but, as subsequently s hown inthetext, the bordering of the square may be 

 affixed at random, i. e. the words left and right or cos and sin may be in- 

 terchanged without leading to any error but of a kind susceptible of imme- 

 diate detection and remedy. 



