TRANSACTIONS OF THE SECTIONS. 3 



In introducing the subject of his paper, Mr. J. T. Graves took occasion to point 

 out that the late Mr. Peter Barlow's valuable work 'On the Theory of Numbers/ pub- 

 lished in 1811, which is the only elemenary text-book of note in our language spe- 

 cially directed to that subject, is not sufficient for the requirements of modern English 

 students. 



¥ 



Two Memoirs. — I. On a Theorem in Combinations. II. On a particular Class 

 of Congruences. By Hbnry M. Jeffery, M.A., Second Master of Pate's 

 Grammar School, Cheltenham. 



9 



I. A Theorem, in Combinations. 



1 . It is proposed to determine the number of combinations of n things 

 taken severally I, 2, 3, ... n together, where there are p of one sort, q of 

 another, r of another, &c. 



We will begin by examining a simple case, where there are three quan- 

 tities, a, b, c. 



The product of the factors 



(l-\-ax+a'Jo^){l + cx), 

 or 



l + {a + c)x-\-(a^+ac)sr-{- a^cx^, 



contains the combinations of the three quantities taken 1, 2, 3 severally 

 together. 



Their numbers in each case are found by equating a, c to unity ; or 



;^Ci = 2; ^0.2=2; 363=!; 

 subject to the above restriction, that two of the three quantities are equal. 



The same process of reasoning is easily extended to the general case, as 

 proposed. 



'I'he product of the factors 



(l+ax + a^x'+ ■\-aPx'P 



■x{\-\-bx+bV+ +J9a;') 



x(1 + cj? + cVH- +c''.c'') 



X 



contains the combinations of the n quantities taken severally 1, 2, 3, .. n 

 together, viz. in the coefficients of j?, x^, x^ . . x^K 



The number of the combinations in each case is found by equating a, b,c . . 

 to unity. 



Hence any particular combination ^C/^ is found by finding the coefficient of 

 that power of x in the expansion of 



{l+x + x^+.. +xP){l+x + ar+ .. +x9)(l+x + x^+ .. +x'') (A) 



whose index is k. 



Or the rule may be otherwise conveniently stated : „C^= the coefficient 

 of x'^ in the expansion of 



l—xP+^ 1— a;9+» l—x''+^ 



1 — X ' 1 — X ' 1 — X 



2. It is important to observe that, subject to these restrictions, 



as is proved by the circumstance, that x and 1 may be interchanged in the 

 above formula (A) without altering its value. 



1* 



