166 
Proceedings of Eoycd Society of Edinburgh. [sess. 
Hence (c) gives, by operating with S./x, S.r, and 
Spn{/T=l . . . . (1) 
1 = nSrcj^T + S/jof/jx Sri^r . . (2) 
— 1 = 7lSp\J/~ 1 (])T + St/) SfJaJ/{l . . (3) 
These preliminaries being settled, we must find the envelope of 
(b) subject to the sole condition (a). We have at once by differen- 
tiation 
S pdf], = 0 , and S dp(cf)r — i/^/x-St^t) = 0 , so that 
xp = cf>r — \]/p$ , ... ( d ) 
Treat this with the three operators used before, and we have 
respectively 
x = S/jLif/fj, Sr<£r . 
Srp = 0 
xSp\J/- 1 p = Spi]/- 1 4 >t + St<£t . 
By means of (5), (3) becomes 
— 1 = 
so that (6) takes the form 
X$p\J/~ 1 p= hS t(£t. . 
7b 
Substitute for \ if/p, in (d) its value in terms of r from (c) ; and x 
becomes, by (4) and (6), a factor of each term ; so that 
p = — nSpfi~ 1 p . <£r - i]/t . . . ( d ) 
Eliminating r between this and (5) we have finally 
S . p(f/ + nSp\J/~ 1 p . </>) -1 p = 0 . 
(Equation (2), above, has not been, so far, required : — but it is 
necessary if we desire to find the values of S/n/7/, and other con- 
nected quantities.) 
It is obvious that, if we had originally eliminated e instead of r), 
we should have obtained the (apparently) different form 
S . p(cj) + mSpcfi ~ 1 p . if/) ~ 1 p — 0 . 
It is an interesting example in the treatment of linear and vector 
functions to transform one of these directly into the other. (Tait’s 
Quaternions , § 183.) 
0 ) 
( 5 ) 
( 6 ) 
( 6 ) 
