568 PROFESSOR KELLAND ON GENERAL DIFFERENTIATION. 



to answer the desired end. We cannot find that M. Liouville has attempted to 

 obtam the differentials of quantities without direct recourse to his complementar}" 

 function ; and consequently, although we acknowledge a degree of improvement 

 on his first essay, we are far from thinking the subject free from objection. We 

 are not aware that M. Liouville has done any thing further in the establishing 

 of the first principles of the science. W^e have three other memoirs by him, two 

 on formulse, by which the subject may be applied, and the third on the mode of 

 effecting a transformation of the independent variable, in none of which does he 

 say a word about the principles. 



About the end of 1837, appeai-ed a paper in the Cambridge Mathematical 

 Magazine, the author of which is ignorant of the greater portion of what has been 

 written on the subject : indeed, he must be supposed to have read M. Liouville's 

 first series of memoirs, and those only. The author professes to have translated 

 a part of these memoirs, altering the parts against which objections had been 

 raised. Independently of the fact, that there are proofs of the limited extent of 

 the author's reading, the mode in which the subject is treated is such as to de- 

 serve a high degree of praise. M. Liouville had left the matter very vague as 

 regards the determination of all differential coefficients, except those which come 

 under a particular /orw. This vagueness is done away with by the author of this 

 paper, who shews how to proceed in all cases of powers of the independent varia- 

 ble. The whole subject, too, is arranged in a logical form, commencing with a 

 generalization of the fundamental formuhe of the science. This, as far as we 

 know, is all that has hitherto been written on the subject, if we may except ge- 

 neralizations of M. Fourier's theorems, and an arrangement of Euler's notion. 

 The subject is indeed in its infancy, but it is to be hoped that it will rapidly grow 

 to a full stature. We venture to express our belief, that the excessive rage for 

 elliptic functions, which has engrossed analysts for the last ten years, will be 

 turned, partially at least, into the channel of general analysis. That we may 

 contribute our part towards effecting this desu*able object, it is our intention to 

 ])resent to this Society two or three memoirs on the subject, endeavouring to place 

 it in so simple a light, that no greater difficulty shall be experienced in appre- 

 ciating the evidence on which it rests, than is attached to common algebra. We 

 hope, too, we shall be enabled to remove the barrier, which, doubtless, has been 

 the real obstacle to its reception, viz. the extreme rimltiplicMtion of cases, which 

 Mr Peacock complains of with justice. To explain what we mean, we may be 

 allowed to observe that, in the present state of the science, the expression for 



\^^ has four different and very dissimilar forms, depending on the signs and 



the relative magnitude of m and n. We shall shew, in the sequel, that one form 

 comprehends them all ; and shall thus be enabled to give to this science its pro- 

 per dignity, by relieving it from the imputation of being subject to a tentative 



