420 MISCELLANEOUS EXAMPLES. 



19. In Art. 275, if p = n 1, we have approximately when 

 x and y are very large 



shew that if q = n 2, we have by continuing the 

 approximation 



y /A\i 5 C 



- = /*i+ -") + -+1 + ... 

 a? VJB/ a; jpt 



^'(/O '"00^ 



where ^=-7- -.; . 



~ f " W 4- f f 



20. If (a, $) be a poiut of the curve <f> (x, y] = through 

 which pass n tangents, shew that the locus of all the 

 tangents at that point is expressed by 



21. Shew that the theorem of Art. 91 will hold even if <j>' (x) 



is infinite when x = a or when x = />. Give a geo- 

 metrical illustration. 



22. Shew that the theorem of Art. 98 will hold even if F' (x) 



or f (x) is infinite when x = a or when x = a + h. 



23. Shew that the formula (3) of Art. 373 will hold provided 



p + 1 is not less than q. 



24. Obtain from (3) of Art. 373 the result 



6}"-* h n+1 F n " (a + 07/) 



1.3.5 ... '(2q+l)\n 



J : PKIXTED BY C. J. CLAY, M.A. AT THE UMVEBSITY PHES3. 



