of Definite Integrals. 439 



This theorem by making m= — \ shews how to determine the 

 quantity of which the definite integral is ./'(i), the differential 

 coefficients are in this case of a negative order — n; that is, 

 they represent the n"' integrals of the quantities under them. 



If we know the first, second, third, &c. definite integrals, 

 we may find the function itself by the following formula. 



Theorem VII. If i - /'. ' + h be the limits of integration, 

 then shall 



these general properties of definite integrals might be much more 

 extended, but the consideration of them would extend this paper 

 to an inconvenient length ; but we may observe that in studying 

 these properties we should divide the functions under the sign 

 of integration into large classes possessed of some common pro- 

 perty, such as vanishing when x = so , &c. : the results do not 

 here possess altogether the generality of the former, but they 

 are more remarkable, and even more useful in analysis. To this 

 part of the subject, however, I shall not at present further allude, 

 but conclude with observing that 1 have subjoined as an illus- 

 tration of the use of the preceding methods,— the general resolution 

 of Riccati's Equation, by means of definite integrals. 



Vul. III. P'lrt III. ' SL 



