Remarks on Several Subjects. 135 
stant) dv=o, and the differential becomes d T=o, there- 
fore the maximum or minimum of u will be obtained by 
puting the differential of T equal to nothing. 
It will sometimes be more simple to put «=yt (instead 
of y=ct) and then we shall have wy" IT’, v=y". T” (T’ 
and ‘T’’ being in general different from the values above 
found) hence, we get as, above 
To ( ue 
n n 
and the maximum or minimum is found, as before, by put- 
ting du=o, dv=o which gives dT=o. 
It may so happen that the proposed problems, without 
any reduction, appear under the form u=o”™. 'T’, v=a". 
T’ ; T’, UT”, being functions of y alone without x. In this 
case no reduction will be necessary, because we have im- 
; re vheed GY 4 ; 3 . 
mediately —_—=__ _= I, whose differential gives dT =o, 
when wis amaximum or minimum, ‘The same thing takes 
place, if u=y" I’; v=y". T’; T’, T”, being functions of 
Sica lols DAS LN ie 
w alone, because they give Ame =T,and the maximum 
(HES {Du : 
or minimum of w is found, as before, by putting dT =o. 
We may observe that generally when dT =o, we shall 
also have d. T?=o, therefore instead of T, we may take 
any power p ofl’, positive or negative ; we may also 
neglect any constant factor a by which T is multiplied, 
since d. aT'=0, gives a. df =o, whence dY'=o. 
All the problems actually solved by professor Fisher, de 
pend on homogeneous functions and can therefore be sol- 
ved by the method just mentioned, putting d! =o ; as we 
shall show by solving a few of his problems. 
Problem'1. Suppose u=ty/yy+ox, v= 3% YL, T=3, 
14159, and let it be required to find the value of w when a 
maximum, for a given value of v. 
The substitution of e=yt, gives u=t. y?. JI +t; v= 
1@.y%.t, Comparing with the above forms u=y". T’, 0 
=y".T", we get m=2 n=3, T=a/1+tt, T’= Jt, and 
4 
if we neglect the constant factors in T = aint becomes 
i} 
T'= 
