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XXXII. Some Account of a Method which may he applied to the 

 same Purposes as Sir Isaac Newton's Method of Fluxions. 

 By Mr. Thomas Tredgold. 



Leti'er II. 



To Mr. Tilloch. 



Sir, — In my first letter (see p. 177 of this Number) on this 

 subject, I explained a method of exhibiting the areas and the 

 lengths of curves by means of progressions. I intend to show 

 how the same principle may be applied to other parts of the 

 method of fluxions ; but, before I proceed further, it will be ne- 

 cessary to show how such progressions may be summed : not that 

 the problem has not been previously tret^ted, but only because I 

 fallow a method of my own, which, perhaps, has advantages not 

 common to other methods. 



Assume a progression of the following form, and range the 

 first differences of the adjoining terms below it : thus 



0+a"+(a + dZ)" + (a + 2i)"+ . . . .{a+\rii-\]d)''-\-{a + md)'. 



a", (a+c/)"-a",(a + 2(Z)"-(a + t^)",..(a + w^)"-(a+[m-l]rf)"- 



Then, it is easily proved that the last term of the progression 

 is equal to the sum of all the differences. Also, if the second 

 differences be taken, the last term of the first differences is equal 

 to the sum of the second differences ; and so of any other order 

 of differences. 



For example : If n = 1, the differences will be a, d, d, d, &c. 

 and if S be taken to represent the sum of these differences, 

 S = a + md. 



Also, if n = 2, then the differences will become 



a, 2ad+d\ 2ad-VZd\ 2arf+ (2m— \)d\ 



and their sum S = {a + mdy. 

 When a = the progression becomes extremely simple, and 

 under this form I intend to show its application. 



Prog. + rf"+(2ci)"+ (3J)"+ {\in-\-]d)'+ (md)". 



Diff^ d\{2d)"~d\{3dy'-{2d", {mdf-{[m-rid)"- 



By means of this progression, any other which increases or 

 decreases by the same law as the differences may be summed. 

 This is effected by comparing the last term of the proposed one 

 with the last difference of this progression : and if both be ex- 

 pressed by the same power or powers of m, with constant quan- 

 tities; it only remains to determine the value of d, which is a 



constant 



