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XXXVI. — On Functions with Recurring Derivatives. By Edward Sang, Esq. 



(Read 5th March 1866.) 



The subject of the following paper is the first proposition in the Calculus of 

 Primaries. The business of that calculus is to discover the relation between the 

 primary variable and its function, when the relation subsisting between the func- 

 tion and its derivative is known. The simplest relationship between two variable 

 quantities is proportionality when they are heterogeneous, or equality, when they 

 are of- one kind ; and the case of proportionality can always, by a change in the 

 unit of measure, be brought to an equality of the representative numbers ; so 

 that our first proposition becomes this : " To investigate the nature of those 

 functions which reappear among their own derivatives." Since this reappearance 

 must necessarily be periodical, I shall use the name Functions with Recurring 

 Derivatives. 



My attention was first drawn to functions of this class by observing that the 

 method of solving Algebraic Equations which I published in 1829, can readily 

 be extended to equations into which a recurring function enters. In that publi- 

 cation an example is given of the solution of an equation of the form 

 ax + b sin x + C = 0, and this example serves as a type for all equations com- 

 posed of an algebraic and a recurring function. 



1. If, after having differentiated a function several times, we come to the 

 function itself as a differential coefficient, it is obvious that the continuation of 

 the process must reproduce the same series of derivatives, and that the group or 

 period must recur again and again. Hence these functions may be arranged in 

 orders, according to the number of terms in the group ; thus the circular func- 

 tions sine and cosine belong to the fourth order. 



2. Hence the sum of all the functions forming a complete period must be a 

 recurring function of the first order. 



3. When the values, corresponding to a given argument, of all the functions 

 of a period are known, their values corresponding to another argument can be 

 computed. 



Let, for example, <pt represent a function of the argument t, such that its third 

 derivative is again <pt ; that is, such that 3t (j)t= cpt ; then must we have 4t (pt — lt <pt, 

 u(pt = 2t<pt, et <pt = 3t (pt, etc. On the supposition that t becomes t + u, the new function 

 becomes, according to Taylor's theorem— 



<P(t + «) = ft + itf j + 2 P*Y72 + if ' ITO + 4 ^ 1.2.3.4 + etC- 

 VOL. XXIV. PART III. 7 C 



