140 Transactions of the Royal Society of South Africa. 
and by using in the concluding equation v, j in place of /x, i with the 
explanation 
y = l, 2, n \ . 
The theorem would then have been satisfactory on the score of accuracy, 
the n'^ given equations being sufficient to ensure the validity of the n^ 
resulting equations. It would still, however, have been lacking on the 
score of deliniteness, because of the failure to indicate how many of the 
given equations were necessary for the validity of any particular one of 
the resulting equations. 
3. Very probably this neglect arose from the fact that the author was 
concerned with a particular application of his theorem, namely, towards 
the generalization of Lagrange's expansion of F(?^) in a series of ascending 
powers of x when 
^l = a^-xf{ll), 
and that the enunciation in question sufficed for the purpose he had in 
view. Probably also this accounts for the care taken to specify in the 
enunciation certain functions as being explicit and certain others implicit. 
4. Gilbert's memoir, however, calls for attention altogether apart from 
these considerations, the proof which he adduces in support of his 
theorem being far from satisfying. What he gives is merely a verifica- 
tion of the three simplest cases, with the second case resting on the first, 
and the third on the first and second. In many instances this might be 
quite convincing as forming the basis of a proof by the method of so-called 
' mathematical induction ' : but here it is not so. 
The purpose of the present note is to effect an improvement under 
both the heads mentioned. 
5. When divested of all superfluities the theorem is : If the differential 
coefficients of y^, j^, y„ with respect to x he iwoportional to the diffe- 
rential coefficients luith respect to I, the common ratio being E, and if the y's 
he also functions of w^, W2, ... w,,, then 
^ \ I ^ VI ii^^y^^ y^^ V" ) I _ ^(^i, Vn, 
I t)(z(;j, ID^, IV„) I ()i I 0(^1, "f^n) ' '^{I'i^i, Q 
In the proof of it we shall take n=--S, not, as will be seen, because the 
method is lacking in generality, but merely in order to save space in 
writing. 
