348 



S.df(Q) = S.iR'QR'R + R"RQR')dQ 



= S.NdQ. 

 Retracing the steps of this process, we see that 



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

 :=S.R Q_R QR" + S.R QR'QR" 



= 2S.f(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 n to be- 

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

 function J" 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 

 ax 



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

 discovered by Fontaine, viz., that if [7 be a homogeneous func- 

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



dU dU dU 



X —r- + y —,— + z —r- + . . .-nU. 

 dx dy 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 vf^ degree, in x, y, and z, the equa- 

 tion of its tangent plane at the point xyz will be 



dU , dU , dU , 



—-- X + — — y + — - z = w const. 

 dx dy dz 



It was from observing the existence of a similar relation 



