Page 88 - Foundations Of Differential Calculus

4. On the Nature of Differentials of Each Order 71
For differentials of the second and higher order, consider the powers of dx;
if the numbers are the same, the expressions are homogeneous.
2 2
3
136. Thus it is clear that the expressions Pd y and Qdy d y are mutually
2
2
2 2
homogeneous. For d y is the square of d y, and since d y is homogeneous
4
2
2 2
with dx , it follows that d y is homogeneous with dx . Thus, since dy
3
3
is homogeneous with dx and d y is homogeneous with dx , we have that
4
3
the product dy d y is homogeneous with dx . From this it follows that
2 2
3
Pd y and Qdy d y are mutually homogeneous, and so their ratio is finite.
Similarly, we gather that the expressions
3 2
5
Pd y and Qd y
2
dx d y dy 2
2 3 5
are homogeneous. If we substitute for dy, d y, d y, and d y the powers of
5
2
3
dx that are homogeneous with them, namely, dx, dx , dx , and dx ,we
3
3
obtain the expressions Pdx and Qdx , which are mutually homogeneous.
137. If after reduction the proposed expressions do not contain the same
powers of dx, then the expressions are not homogeneous, nor is their ratio
finite. In this case one will be either infinitely greater or infinitely less than
3
the other, and so one will vanish with respect to the other. Thus Pd y/dx 2
2 2
to Qd y /dy has a ratio infinitely large. The former reduces to Pdx and
3
the latter to Qdx . It follows that the latter will vanish when compared
to the former. For this reason, if in some calculation the sum of these two
terms occurs
3 2 2
Pd y + Qd y ,
dx 2 dy
the second term, compared to the first, can safely be eliminated, and only
3
2
the first term Pd y/dx is kept in the calculation. There is a perfect ratio
of equality between the expressions
3 2 2 3
Pd y Qd y Pd y
dx 2 + dy and dx 2 ,
since when we express the ratio, we obtain
2 2 2
2 2 2
Qdx d y Qdx d y
1+ =1, because =0.
3
3
Pdy d y Pdy d y
In this way differential expressions can sometimes be wonderfully reduced.
138. In differential calculus rules are given by means of which the first
differential of a given quantity can be found. Since second differentials are
obtained by differentiating first differentials, third differentials by the same
operation on seconds, and so forth, the next one from the one just found,