AS COMMONLY INVESTIGATED. 231 



the co-ordinates, and may occur in a curve having a continuity of the first de- 

 gree; the second and third, which are the only points of reflection usually 



e 



^ ' 



A. '^ A< 



given, appear in continuities of the second order; whilst the fourth, which, 

 viewed independently of the arbitrary arrangements of analysis, seems to present 

 no greater breach of continuity than the second or third, is yet, from admitting 

 two tangents at A, not regarded as a continuity by our analysts, and can only 

 appear in those transcendants which we have described as possessing continu- 

 ities of the third order.* 



This last division falls, therefore, under the class which mathematicians 

 have termed discontinuous, and which we propose to designate as continuities 

 of orders lower than the second. With regard to such functions, which phy- 

 sical problems every where present us, and which now act so important a part 

 in mathematics, some confusion appears to exist. Most authors, and Fourier 

 and PoissoN in particular, have endeavoured to reduce their analytical expres- 

 sions to that connected series of arrangements which I have described as con- 

 stituting the arranged functions of numerical logic, and which is generally 

 known as "the calculus." 



Mr. Peacock, in a report to which I have frequently alluded, proposes, as, 

 indeed, had been informally done before, to introduce a sign of discontinuity 

 ^D''; which amounts to an agreement that in such an expression as 



y = A log X -f- B sin X (6) 



X X 



the coefficients A and B shall not be absolutely constant, but that A shall be 



XI X 



unity and B zero between the limits x = — infin. and x = n; whilst, on the 



X 



contrary, these coefficients are to interchange those values from x = n to 

 X = + infinity. 



This method of proceeding, which has been often used, differs from the ver- 

 bal statement of the proposition in no other way than as agreeing upon some 

 common symbol which expresses that statement in the shortest manner. It 

 affi3rds, therefore, no assistance towards reducing y to the arranged functions 

 of the science, and, consequently, never leads to the remarkable expressions 



* The branches are supposed to terminate at A, otherwise this would merely be a multiple point, 

 and what is said above would not be true. 



