520 MR EDWARD SANG ON THE THIRD CO-ORDINATE BRANCH 



daily occurrence, is arrested, not by our ignorance of the mechanical principles in- 

 volved, but by the imperfection of our attainments in arithmetic. 



The integral calculus may almost be described as a collection of artifices by 

 which a great variety of proposed differentials may be transformed into other 

 differentials of which the integrals are known, not by integration proper, but by 

 previous differentation. These artifices must necessarily bring us to known 

 functions, or to combinations of known functions. Now the number of simple 

 functions employed in the calculus is very limited ; the usual enumeration gives 

 us the potence or algebraic function, the exponential, the logarithmic function, the 

 sine, and the cosine, in all five ; and even of these five two must be removed, 

 because the logarithmic and exponential functions are converse to each other, 

 while the cosine is only a variety of the sine ; and thus we have but three kinds 

 of simple relationship from which we seek to compound all others. 



When we contrast this paucity of material with the multifariousness of the 

 work to be accomplished, we need not be surprised that so much remains to be 

 done ; it is rather matter of astonishment that so much has been performed. The 

 mutual relations of physical phenomena are too various, too complex, to be re- 

 presented by any combinations of such a small number of functions ; and our 

 hopes of passing beyond the present limits of applicate analysis must be founded 

 on an extension of the groundwork, and on an enlargement of our plan of 

 operation. 



In the calculus of primaries, the first problem which presents itself, that in 

 which the relation of the function to its derivative is of the very simplest kind, 

 is to investigate the nature of those functions which are equal to their derivatives. 

 The solution of this problem leads us to classes of functions of which the ex- 

 ponential and the circular are cases ; that is to say, the whole of the actual cal- 

 culus is only the exposition of part of the simplest proposition in the theory of 

 primaries. It is, then, not unreasonable to hope that the farther cultivation 

 of this theory may enable us to resolve problems which have hitherto resisted 

 all our efforts. 



In arranging the parts of this theory, we may place under one general head 

 all those cases in which the relation between the function and one of its deriva- 

 tives is given, and we may call problems belonging to this head pure problems. 

 Thus, when we investigate the motion of a body which is drawn toward a fixed 

 point by an attraction depending on the distance, the relation between the func- 

 tion and its second derivative is given; or, if we be inquiring into the form 

 assumed by an elastic plate, of uniform breadth, when vibrating, the relation 

 between the ordinate and its fourth derivative is prescribed ; these investigations 

 lead to pure problems. 



When two or more derivatives are combined in the relationship, we may give 

 the name mixed problems to those which result. For example, the motion of a 



