63 
and thus, although he has not determined the order of T, nor 
assigned the form of his substitutions when Jc > 1 (for he has 
not at all inquired into the degree of his unknown functions 
of i, the (r + 1)"‘ variable which has k values), he conceives 
that he has exactly given the superior multiple of any group T, 
Avhich has a set of similar substitutions PjPj. . . the whole of 
which form, with their derivatives, an intransitive group 2. 
He has, indeed, assigned the superior limit of many such 
wove?i grouped groups T ; but there is a vast number of them 
whose order is out of his limit. Take, for example, the woven 
grouped group of my theorem L (Art. 51), which is made by 
writing under 12345 the long-known group of 60 made with 
those elements, parallel to the same group of 60 made with 
67890. Weaving the two sub-groups of 60, and then trans- 
posing them, or, which is the same thing, multiplying by 
6789012345, we have a transitive group T of 60 , 60’2. If we 
collect the 48 substitutions similar to 1234578906, i.e. to 
2345167890, of the fifth order, these determine an intransitive 
2, containing the five powers of 2345178906, and no 
regular substitutions hut of the fifth order. No other 
intransitive 2 is thus constructive. Therefore n = 5 
is the order of the sous-groupe minimum, M = 2'S 1 ' 1 = hi rp , 
and the order of T ought, by M. Jordan’s result, to divide 
1 *2* { 5*4‘ } 2 = 800. But 60T20 is no divisor of this number. 
We can also construct such a group T of the order 120'240, 
which is no divisor of 800. 
For another example, write under 123456 the group of 
60 positive substitutions, selected from the group of 120 of my 
article 65, parallel to the same group made with 7890«5. 
We have by weaving and grouping, as before, a transitive 
group T of the order 60T20, in which, if we collect all the 
substitutions similar to 265341 7890a5 of the third order, 
these determine a derivative intransitive 2, the only one 
constructive, which has regular substitutions of the third 
order, and of no other. Then n = 3 is the order of the sous~ 
