Mr. Robertson’s new Demonstration 
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if we put s= the number of times each quantity is found in a 
coefficient, n= the number of quantities a , b, c, &c. used in 
producing the equation, and p= the number of quantities in 
each member ; then as a is found s times in this coefficient, 6 
is found s times in this coefficient, &c. the number of quan- 
tities in this coefficient, with their repetitions, will be sy.n; and 
as p expresses the number of quantities requisite for each 
member, the number of members in the coefficient will be ~ • 
P 
Thus, for the sake of illustration, if we limit the above no- 
tation to the second term of the equation of five dimensions, 
5=1, as each of the quantities a , b, c, &c. is found once in the 
whole coefficient of x 4 ; p= i, as each member consists of one 
quantity, and n=5, as a, b, c, d , e are used in producing the 
equation. Consequently ~ =5. If we limit the above nota- 
tion to the third term of the same equation, 5=4, p= 2, and 
11=5, and therefore — =10. If we limit the above notation 
to the fourth term of the same equation, 5=6, p=s, and n=5, 
and — =10. If we limit the above notation to the fifth term 
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of the same equation, 5=4, ^=4, and n=5, and — =5. 
11. Using the same notation, we can by the last two 
articles, calculate the number of members in the next coeffi- 
cient after that whose number of members is y. For as — 
expresses the number of members in the above mentioned 
coefficient, and 5 the number of times each quantity is found 
in ^ — —5= the number of times each is not found in it. By 
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the 9th article therefore, a will be found y — s times, b will 
be found — —5 times, &c. in the next coefficient, and there- 
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fore — — 5 x n = = the number of quantities, with. 
