364 macfarlane: algebra of physics 



Since e b ^ /2 = 1 + b^ /2 + | f + £ /3 37r/2 + • • • • 

 and ^ = 1 + c T 7r/2 + £ 7^ + £ 7 37r/2 + • • • 



„7i72 7T/2 e& /»3 



e 6/3 *"' = l + C7 ^+^ + |L y W2 + 



= 1 + W' 2 + Ct" 72 + \ I & 2 /3 T + 2bc^ /2 y w/2 + cV } 



+ 1 1 6 3 /3 37r/2 + 3& 2 c /3V /2 + 36c 2 ^ 7 ff + c 3 7 37r/2 } 

 + etc. 



The question now is: Is the quadratic expression within the 

 brackets the square of b^ /2 + cy^ 2 ? Hamilton answered in 

 the negative, because it does not conform to the formula 



(A + BY = (A + B) (A + B) = A 2 + AB + £.4 + B\ 



I answer in the affirmative, because it conforms to the formula 



(A + BY = A 2 + 2AJ5 + B\ 



The latter is the square of a succession of vectors A and B, 

 the former is merely the square of the resultant of A and B. 

 Observe also that the above method of multiplying gives only one 

 bc^ /2 7 7r/2 , and that the 2 is introduced to compensate for the 

 factorial 2! placed outside; the multiplication gives no term in 

 b c 7 7r/2 $* 12 as is given by Hamilton's conjugate square. In a 

 similar manner the terms within the next bracket give 

 \b(i w/2 + C7 ?r/2 } 3 ; hence the theorem is proved. 



In this way the doctrine of successive vectors and of direct 

 powers was introduced into space-analysis. The truth of the 



