270 Mr. Challis on the Theori/ of the 



series be summed to infinity, this result is obtained. But the 

 objection is, that a mode of summing a converging series is ap- 

 plied to one which is not convergent. The only legitimate method 

 is to sum the series to m terms, and to find what the sum be- 

 comes when m is infinite. Lagrange does this ; he finds the sum 



, , con m9 — cos {m + i) 6 1 , ^.u * 4^u ^ 4. ^ 



to be — 37 -, and says, that the first term 



2(1 — COS u j 2i 



disappears when m is infinite, because the 1 may be neglected 

 in comparison of m. But it cannot be admitted that two arcs, 

 however great, which differ by a quantity Q, have the same 

 cosines independently of the value of Q. The fallacy of the rea- 

 son assigned for neglecting i, will be apparent, by putting the 

 sum of the series under this other form, 



JK + 1 „ . mQ 

 cos Q sin — 



2 2 



Q 



cos - 



2 



which does not give the same result as before, when 1 is neglected 

 in comparison of m. I have adverted to this error, because in 

 consequence of it, Lagrange exhibits to view a discontinuous 

 function, the possibility of doing which, may well be called in 

 question. It is not necessary to enquire how tli«> reasoning may 

 be conducted, if this step be corrected, because the second 

 Researches are in principle the .same as the first, and are 

 not liable to a .similar objection. In these he has elaborately, 

 yet .strictly shewn, as far as I have been able to follow 

 the reasoning, that the motions he is in search of, are not 

 subject to any law of continuity: — that the motions, for instance, 

 at a given' in.stant, in a column of fluid stretching between two 

 given points, cannot be given generally by any known line or 

 function. He supposes, therefore, that they will be given, by a new 

 set of functions, neither algebraical, transcendental, nor mechanical, 



