270 First Principles of the Differential Calculus. 
dependent of h, also that B is a function of z and h, such that it 
=0 when h=0, and that yh is a function of 4 independent of «, 
and such that it =0, when h=0; for according to these supposi- 
tions when h=0, (1) becomes-identically gxr=9z, as it (evident- 
ly) ought to be. 
_ Since A is arbitrary, we may put 2h for h in (1), and if we use 
B’ to denote the value of B when h is changed to 2h, (so that 
B’ is the same function of x and 2h that Bis of x and h,) (1) 
becomes. o(4+2h)=9r+(A+B’)y(2h), (1’); also since x is ar- 
bitrary we may put #+A for x in (1), and if we denote the in- 
erements of A and B (arising from the substitution of r+A for z 
in A and B, which are supposed to be functions of z,) by 4A and 
4B; it becomes 9(@+2h)=9(4+h)+(A+B)yh+(4A+4B)yh, 
or substituting the value of ¢(z+h) from (1), 9(#+2h)=9r+ 
(A+B)2yh+(4A+4B)vh, (1); and subtracting (1”) from (1’), 
we get A[y(2h) —2vh] + B’y(2h) —2Byh —[4A + AB]vh = 0, 
which must be an identical equation ; .*. since A is independent 
of h, and B’, B, 4A, 4B, are not independent of it, (since each 
of them =0 when A=0,) we must have 4(2h)—2vh=0 or 
y(2h)=2yh, and since his indeterminate y=1, .°. 2h=2h, an 
identical equation, and A is arbitrary, as it ought to be; hence 
the equation is easily reduced to 2(B/—B)—sA—4B= 0, (2), 
which must be satisfied so as to be an identical equation. — 
Since y=1, (1) becomes o(a-+h)=92+(A+B)i=o2+Ah+ 
Bh, which thal that Ah+Bh is the increment of yz arising 
from the substitution of 7+-A for x, .". we may denote this inere- 
ment by 4a, and shall have 49r=Ah-+Bh, (3), so that (1) be- 
comes 9(2+-h)=9r-+ 49x, (4). id 
First Principles of the Differential Calculus. 
We may consider / as an increment of ms on denote it by dn, 
and (3) becomes 49¢=A4c+B4s, (3”), o1 =A+B, (3”)5 
which must manifestly be an identical seni: and be satisfied 
so as to leave 4x indeterminate; .:.since A is independent of 42, 
(or h,) the first member of the equation must be considered as 
having a ric which is independent of eat .". if we denote this 
term ws “ , We get ee oA, (4’), or dna = dx=Adz, (493 
where z is called the independent variable, prvieab of 2; 
