AS COMMONLY INVESTIGATED. 219 



such as admit of only one form, and that always give the same terms when the 

 function operated upon is varied in any way that does not affect its value. 



X 12 



Hence, if such a form can be found for v — the general term of v, v, v; as, 

 within specified limits of some of the variables that enter u — shall make any 

 one of the terms in (2) greater than all which follow — including R — the 

 functions 



0000 1100 11 22 



U V, U V + U V, U V + U V + U V, &c., 



will be osculates of the more complicated, and less known, function u, or, in 

 other words, will agree with it more nearly than any other functions of the 

 same peculiar form and simplicity can agree. 



The success of the method obviously depends on our obtaining known and 

 manageable functions for the derived terms, or, rather, for those partial sums 

 of them which we have above enumerated. 



The great importance of such osculates will be felt when we consider that 

 not only tangents, tangent planes, circles of curvature, motions in these lines 

 and planes, and other similar geometrical and dynamical elements, are merely 

 osculates to the function they are intended to trace, but that every physical 

 hypothesis stands in the same relation to the function which it represents; as 

 may be well perceived in the Newtonian law of gravity, which is merely re- 

 garded as a convenient and finite osculate to the law of nature. 



Recent analysts, and especially Lagrange and Fourier, have greatly added 

 to this method of investigation, by the introduction of those very simple tran- 

 scendental osculates which give finite osculations with any required portion of 

 a function., and which thus enable us to separate the parts that belong to the 

 problem in hand from those which would merely embarrass the investigation. 



Having thus pointed out the manner in which the differential calculus pro- 

 ceeds, it may not be amiss to notice that of its converse, whereby we shall 

 throw farther light upon the nature of the developments we are considering. 

 These two branches of numerical logic are, indeed, so imperfectly explained 

 in our elementary works, and have been viewed by writers of the eminence of 

 Cauchy and Mr. Peacock in a manner so much at variance with what I re- 

 gard as their true principles, that I found this little preliminary sketch essen- 

 tial to the clear understanding of the principal subject of this paper. 



