480 Proceedings of Royal Society of Edinburgh. [july 18, 
The first row of the permutations involving two letters is got by 
taking the first letter of the previous row and annexing each of the 
others to it in succession and in the orde** of their occurrence ; the 
second row is got in like manner from the second letter ; and so on. 
Similarly the first row of permutations involving three letters is got 
from afi the first obtained permutation of two letters, the second 
row from ay the next obtained permutation of two letters, and 
so on.* 
The second problem (and on this occasion actually so designated) 
is somewhat quaint in its indefiniteness, viz., to prefix to each per- 
mutation the sign + or the sign -- , so that the sum of all the 
permutations involving the same number of letters (> 1 ) may 
vanish (“ Singulis Variationibus, omissis repetitionibus, signa + et 
— it a praefigere, ut summa secundoe et cujuslibet classis insequentis 
evanescat”). There is no indefiniteness or multiplicity about the 
solution, which in substance is : — Make the permutations in every 
row of the preceding table alternately + and - , the first sign of 
all being + , and the first permutation of every other row having 
the same sign as the permutation from which it was derived. In 
this way the table becomes 
+ a, - /3, + 7 , - 5 } 
+ a/3, — ay, + a5 ^ 
-/3a, + &y, - /35 1 
* + ya, - 7 / 3 , +75 | 
- 5a, 4- 5/8, - 57 J 
4- a/ 87 , — a/38 1 
— 07/8, + a7^ 
4-a5/3, — a 57 
— / 3 a 7 , + j8a5 
+ / 87 a, - / 87 s 
— /85a, 4-/857 
4"7a/8, — 7a5 
~ 7 / 3 a, 4-7/85 
4- 75a, — 75^3 
— 5 a/3, 4- 5a7 
4- 5 /8a, — 5/87 
— 57 a, 4 - 57 IS , 
* It will be seen that the order in which the permutations come to hand in 
this process of tabulation is the order in which they would be arranged accord- 
ing to magnitude if each permutation were viewed as a number of which 
a, / 8 , 7 , 5 were the digits, a being < /3 < 7 < 5 (“ ordo lexicographicus,” “ lexi- 
cographische Anordnung” of Ilindenburg). 
