352 MR. WARREN ON GEOMETRICAL REPRESENTATION 



For this may be proved nearly in the same manner as the preceding article. 

 40. Let u = i^\^, where x and m are any quantities whatever, and let 



m 

 remain constant whilst x and u vary, 



. m — 1 



Then ^ = m [^) 

 For, since u = /xV 



u 



= m a/ 



1 du 

 dx 



du 



'. (by Art. 4.) - . 5^ = m . -, 



dx X 





= m 



X 



m- 1 



= "■© 



4 1 . Let X = 1 + a?, and let z be inclined to unity at an angle = 6,0 being 

 positive and less than c, and let x be in length less than imity, and let m be 

 any quantity whatever ; 



then /js\"' = (^Y ' 1 ^ + w ■^ + ^ 'i^ g ^^ + &c. i , if ^ be less than -^ 

 = / 1 X*" . -j 1 + m a? + ^'^^ x^ + &c. > , if^ be greater than 

 For, first, let 6 be less than ^ , 

 and let ^zV' = A + Ba: + C/ + &c., 

 then diflFerentiating 



if) 



3c 



m/zV"' =B + 2C*+ &c.. 



w» — 2 



m.m- I /zy" = 2C + &c., 

 &C. = &C. ; 



now let X = 0, 



