TRANSACTIONS OP THE SECTIONS. 13 



Mathematics. 



On the Contact of Surfaces of the Second Order ivith other Surfaces. 

 By Prof. Clifford, M.A. 



New Improvements in Approximating more rapidly than usual to Square, Cube, 

 and other Boots of a given Number N. By Matthew Collins, A.B. Dublin. 



On Square Soots. 

 It 19 plain that (N^— a)™, where m is a positive integer and N and a any given 

 numbers, has always, when expanded by the binomial theorem, the form AN^— B, 

 since (Ni)^=N.NJ, (Ni)*=NS (Ni)'=N^ N^ &c. 



Now, by one or two trials or guesses a can be taken so near N^ that N= — « shall hs 

 a small fraction /<i, or even <Vo; and then (N*-«)"' =/■'", i.e. =AN*-B, which 

 gives ^ „, j3 



N^"= — ^ , .-. very nearly = ^, 



especially when m is a large integer whose greatness plainly increases A and B, 



N — a^N— rt^ 

 but diminishes/'", where/=N'' -a,.: =-[^t— . ■■. = -^^ nearly, so that /, when 



2a 2a 



Mv. gr. m=3 gives 



^ , (3N+o=)«+/' ,, 3N+a2 /3 



^'= 3a^+N ^^'^'^^^y' •'• = 3Hq:N«+4N '''^'^ ^'^^'^^ 

 (since a-=N nearly), 



positive, is < ^ " or -— , where D=N— a*. 



_3N+fflS 



a nearly ..... (A) 



3a^ + N 



as -^ must necessarily be very small indeed. Now this plainly agrees with Dr. 

 4N ' J 



Hutton's elegant formula (D), given further on, for approximating to N" when ra= 2. 

 But we can approximate to N^ still more rapidly than by Dr. Hutton's rule ; 

 for by taking m = 5, we find 



._ a(aM-10Na-+ 5N-)+/^ ^^^^^ 

 ^ ~ N^+10Na2+5fl^ ^' 



5(N+a^)^-(2a^)' J__^a'+f very nearly (since a2=N nearly), 



5(N+a7-'-(2N)^ ^+(4Nj2 "ry j j v 



, 5(N+a^)^-(2a^)^ ,, „ , /rn 



Now this last neiv and elegant formula approximates to N^ much more closely than 

 the above-mentioned formula of Dr. Hutton, since the eiTor or supplementary term 



here omitted, viz. -^ = 4^. • -j^, is obviously much less than ^ , its value or 

 ' IGPn-' 4N 4iN 4JN 



amount when Dr. Iliitton's rule is used. 



Kv. gr. To find VS. Let us take a =2, and as N is here =3, we find by our 



new formula 



1872. 3 



