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Integration of Differential Equations. 273 
_ “Tshall not, however, here prosecute this subject. I will only remark, 
that fundamental ideas, as we view them, are not only not innate, in any 
usual or useful sense, but they are not necessarily ultimate elements of 
our knowledge. They are the results of our analysis so far as we have 
yet prosecuted it; but they may themselves subsequently be analyzed. 
It may hereafter appear, that what we have treated as different funda- 
mental ideas have, in fact, a connexion, at some point below the struc- 
ture which we erect upon them. For instance, we treat of the mechan- 
ical ideas of force, matter, and the like, as distinct from the idea of 
substance. Yet the principle of measuring the quantity of matter by 
its weight, which we have deduced from mechanical ideas, is applied 
to determine the substances which enter into the composition of bodies. 
The idea of substance supplies the axiom, that the whole quantity of 
matter of a compound body is equal to the sum of the quantities of 
matter of its elements. ‘The mechanical ideas of force and matter lead 
us to infer that the quantity both of the whole and its parts must be 
measured by their weights. Substance may, for some purposes, be 
described as that to which properties belong; matter in like manner 
may be described as that which resists force. The former involves the 
idea of permanent being; the latter the idea of causation. There may 
be some elevated point of view from which these ideas may be seen to 
run together. But even if this be so, it will by no means affect the 
validity of reasonings founded upon these notions, when duly deter- 
mined and developed. If we once adopt a view of the nature of know- 
. ledge which makes necessary truth possible at all, we need be little 
embarrassed by finding how closely connected different necessary tr uths 
are ; and how often, in exploring towards their roots, different branches 
appear to spring from the same stem. - WHEWELL.” 
Grange, August 31, 1840. 
Arr. VIIL—Integration of a particular kind of Differential 
Equations of the second order ; by Prof. Taropore Strona. 
. : d?y 
THe equations which we propose to integrate, are Fo + 
: d?y | 2pq—q+ldy 
dtu ds 
~q?a?2b2y2%-2y=0, (2), in which y and u are the only variable 
quantities, « being considered as the independent variable, whose 
differential (denoted by dw) is supposed to be constant or invarla- 
ble, and y is supposed to be a function of u. ; 
Vol. xu, No. 2.—Jan.-March, 1842. 35 
29 rt eo Wy 4 grardtutt*y=0, (1), and 
