270 First Principles of the Differential Calculus. 
dependent of h, also that B is a function of x and h, such that it 
=0 when h=0, and that yh is a function of h independent of z, 
and such that it =0, when 4=0; for according to these supposi- 
tions when h=0, (1) becomes identically yr=pz, as it (evident- 
ly) ought to be. 
Since / is arbitrary, we may put 2/ 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 z and 2A that Bis of x and h,) (1) 
becomes 9(4+2h)=9r7+(A+B’)v(2h), (1’); also since z is ar- 
bitrary we may put +A for x in (1), and if we denote the in- 
crements of A and B (arising from the substitution of c-+-A for z 
in A and B, which are supposed to be functions of z,) by 4A and 
4B, it becomes 9(7+2h)=9(4+h)+(A+B)yh+(sA+4B)vh, 
or substituting the value of 9(z-+-h) from (1), 9(c-+2h)=9r-+ 
(A+B)2yh+(4A+4B)yvA, (1); and subtracting (1) from (1’), 
we get A[y(2h) —2vh] + B’y(2h) —2Byh—[4A + 4Blyh = 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 h=0,) we must have 4#(2h)—2yh=0 or 
y(2h)=2wh, and since # is 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)—4A—4B=0, (2), 
which must be satisfied so as to be an identical equation. 
Since y=1, (1) becomes ¢(x+h)=er+(A+B)h=or+Ah+ 
Bh, which shows that Ah+Bh is the increment of gx arising 
from the substitution of x-+-h for x, .°. we may denote this incre- 
ment by 4x, and shall have 4gr=Ah-+Bh, (3), so that (1) be- 
comes 9(2-+-+h)=9r+ 4x, (4). 
First Principles of the Differential Calculus. 
We may consider / as an increment of rv, and denote it by Jz, 
A 
and (3) becomes 47z=Adz+Ba4e, (3/), or = A+B, (3”), 
which must manifestly be an identical equation, and be satisfied 
so as to leave dx indeterminate; .:. since A is independent of 4z, 
(or h,) the first member of the equation must be considered as 
having a term which is independent of Jz, .°. if we denote this 
d d dy 
term by s, we get 7A, (4’), or dye = . dz = Adz, (4”) ; 
where z is called the independent variable, yz a function of 2, 
