352 
PROFESSOR DONKIN ON THE 
Appendix B. 
On the subject of the transformation of elements, the following additional remarks 
will hardly be superfluous. Suppose Q is originally a function of the elements a, h, c, 
See. with t ; and let a, (3, y, See. be other quantities connected with a, h, c, Sec., by 
equations such as 
da=Adci-\-Bdj5-\-Cd'y-{- (a.) 
where A, B, C, ... K are given functions of a, (3, y, ^ ; or by equations such as 
da — A ^dQ--\~^ ^dh-\~ . • • ~\~^A^dt , (®^*) 
where A^ &c. are given functions of a, h, , f. In either case, if each of the equa- 
tions be integrable per se, we may consider a, 6, c, .... as functions of a, (3, y, ...., ^ ; 
and such equations as 
da. 
.dO, j^dn 
=A*+B^ + - 
db 
(Q.) 
are both significant and true. 
But if the expressions on the right of the equations («.) be not differentials per se, 
dDj 
the equations (Q.) are either unmeaning or untrue. For the symbol implies one 
of two things; either, that Q is expressed in terms of a, |3, ... ^ without arbitrary 
constants (i. e. that the transformation of Q can be actually effected without inte- 
grating the differential equations of the problem), which is manifestly impossible, 
unless (a.), &c. be integrable per se ; or else, that the differential equations are to be 
conceived to have been completely integrated, so that a, h, Sec., and consequently 
Ap Bp &c., are known as Junctions of t and arbitrary constants, whereby the right- 
hand side of (a.) becomes an explicit function oft (and arbitrary constants), so that 
a, (3, &c. may by integration be expressed in the same way, and, by means of (a.), 
a, b, &c. may be similarly expressed, and finally, by algebraical elimination, a, b, &c. 
become functions of a, /3, &c., t, and arbitrary constants. On this supposition, ^ 
has a meaning, but the equation (Q.) is untrue-, for we must have 
da 
dfl da dn db 
da da'db da~^‘”'’ 
and it is manifestly not true that ^=A, Sec. in this case, because the equation 
da — Aflfes — |— Bc(3 -|- . . . — l\.dt, 
not being integrable per se, only subsists for those variations of a, (3, Sec. which 
actually take place during the instant dt ; whereas the equation 
rfa=|&+|rf/3+...+§* 
subsists for arbitrary variations of all the variables. This view of the subject entirely 
