1. (1) We have, above, assigned an intelligible meaning to this expression if the body be considered as a member of a system. But, taken absolutely, it has no meaning.

2. (2) If our shot, now considered separately, meet directly a target of mass M′ and there be inelastic collision, and if the relative velocity be U, then the velocity u′ of the shot relatively to the c.m. of this new system will be , and the velocity V′ of the target relatively to this c.m. will be .

The total internal energy of this system as measured by the heat given out will be , and of this the shot may with some reason be said to contribute the first term, and the target the second, in virtue of its K.E. [see VII., (3) and (4)].

The following presentation of the fundamental theorem of The Integral Calculus has been evolved as the result of attempts to put § 31 of Whittaker’s “Analysis” before a University class. It is to be regarded as the first application of the Infinite Sequence Theory which has been the subject of articles in recent numbers of The Mathematical Gazette. It is in the details, making for stricter logical accuracy, that what is new will be found.

Theorem. If a he any positive quantity not equal to 1, and x, y, z be any three rational quantities in descending order of magnitude, then