M. Poiusot on the Percussion of Bodies. 251 



whence, putting for i and d—i the preceding values, we deduce 



d^ 



Oi*+M)K2=M 



("■■*# 



It" herein we suppose the mass /* to increase from zero to infinity, 

 the moment of inertia will clearly increase from its least value 

 MD^ to its greatest M(D^ + (/^) ; so that making /x = ao in order 

 to pass to the mathematical hypothesis of a fixed point in the 

 body M, we have 



(/i + M)K2=M(D2 + f/2), 



which is precisely the value of the moment of inertia of the 

 simple body M with respect to the fixed point I. 



7. The moment of inertia of the system having, therefore, a 

 finite value, it is evident that if we represent the same in the 

 ordinary manner by the product (/i + MjK^, the line K or arm 

 of inertia ought to be regarded as zero, the mass (/t + M) being 

 infinite. Nevertheless it is well to notice that this infinitesimal 

 line K is infinitely great when compared Vvith the distance i from 

 the point I to the centre of gravity g of the system, just as the 

 sine of an infinitesimal angle is infinite with respect to its versed 

 sine. In fact, if we compare the expression for K-, which is 



K-^*l['''+»<l+f)] 



with that for i'^, which is 



we find 



which, on making yit= CO , gives -:2=^ > ^"^i^ shows that K is 



infinite times greater than i. 



On the other hand, it will be seen that the quantity -^, which 



in geometry representa a line, corresponds here to a finite line /, 

 and not to an infinite one. For on multiplying both sides of the 

 preceding ccpiation by i, and in the second mcuibcr replacing i 



by its value d — -^., we find 



