348 



S-d/(Q) = S.{R'QR"R + R'RQR)dQ 

 = S.NdQ. 

 Retracing the steps of this process, we see that 



S.NQ= S.{R'QR"R + R'RQR) Q 

 = S.RQ.R'QR"+ S.RQRQR" 

 = 25./(Q). 



And the proof would hold equally good if/( Q) became the sum 

 of any number of terms, all of the same form as RQR'QR". 

 The theorem is therefore proved for the most general homo- 

 geneous function of the second degree of Q. The nature of 

 the proof remaining quite unaltered when we suppose ?i to be- 

 come any other positive integer; and, moreover, conceive the 

 function /" to depend upon any number of variables, it seems 

 unnecessary to occupy space with the fuller statement of it. 



The equation (1) is an extension to the calculus of quater- 

 nions of the ordinary algebraic equation, 



d 



x-r-x" = nx" ; 

 ax 



and equation (2) is an extension of the more general theorem 

 discovered by Fontaine, viz., that if L'^ be a homogeneous func- 

 tion of the n"^ degree of any number of variables, x, y, z, &c., 



dU dU dU 



X -5— + y — -- + z —r- + . . . = nU. 

 ax ay dz 



By means of this latter theorem it is proved that if a sur- 

 face be represented by an equation U = const., in which U is 

 homogeneous, and of the k* degree, in x, y, and z^ the equa- 

 tion of its tangent plane at the point xyz will be 



dU , dU , dU , 



—T- X + — — y + -7- z = wconst. 

 ax ay dz 



It was from observing the existence of a similar relation 



