Integration of Differential Equations. 273 



" I shall 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 kno-.vledge. 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 ti'eat 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. Ther^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 truths 

 are ; and how often, in exploring towards their roots, different branches 

 appear to spring from the same stem. W, Whewell." 



Grange, August 31, 1840. 



Art. VIII. — Integration of a particular kind of Differential 

 Equations of the second order ; by Prof. Theodore Strong. 



d^y 



The equations which we propose to integrate, are ^— j + 



2vq-q + l dy , ^ ..V -, (l^y 2pq-q+ldy 



--q-a^b"u-^~-y=-0, (2), in which y and u are the only variable 

 quantities, u being considered as the independent variable, whose 

 differential (denoted by dii) is supposed to be constant or invaria- 

 ble, and y is supposed to be a function of u. 



Vol. XLii, No. 2.— Jan.-March, 1842. 35 



