macfarlane: algebra of physics 365 



Binomial Theorem, at least for n being a positive integer, became 

 evident, namely, 



(iA + iBY = (iA)» + n (iA)"- 1 (iB) + -y^ (iA)—' (iB)' 



+ etc. 



where iA and iB denote two successive logarithmic vectors aa v/i 

 and 6j3 , and it was inferred that the same would apply to a 

 sum of successive simple vectors A + B where A = aa and B 

 = bp. 



From the Binomial Theorem, the Multinomial Theorem follows ; 

 for example: 



(A + B + CY = A 2 + B 2 + C 2 + 2A£ + 2AC + 2£C; where 

 it is to be noted that the term which occurs is 2 AC not 2CA. 



It was shown that the sum of successive 

 vectors A -f- B — A does not reduce to B, 

 but is represented by the three sides of a par- 

 allelogram (fig. 3) 



On the imaginary of algebra. Proc. 

 A.A.A.S., 41 : 33-55. This paper contains 

 the extension of the prosthaphaerisis theo- 

 rems of plane trigonometry to spherical trig- /wa, 3. 

 onometry; also an investigation of the loga- 

 rithmic circular spiral e 6/3 ", where w is the constant angle between 

 the radius vector and the tangent. The spiral is equivalent to 

 exp (b cos co + b sin w . (3 n/2 ), = exp b cos co exp b sin co . I3 W/2 . 



The expression, complementary to that for a circular angle, 

 was sought for a hyperbolic angle in space, in the simplest case 

 where the hyperbola is equilateral. The difficulty lay in the 

 circumstance that the logarithm of the hyperbolic /3 1 seemed to be 

 (3 X , whereas fi w/2 seemed to be needed to express the rectangular 

 components. The difficulty was then only partially solved. 



Definitions of the trigonometric functions. Read before the 

 Mathematical Congress at Chicago, August 22, 1893. Separately 

 printed. This paper treats of the triangular functions, the cir- 

 cular ratios, the equilateral hyperbolic ratios, the elliptic ratios, 

 and the complex ratios. 



