28 EE. B. Wilson, 
it follows that 
ce—pp—7) —--— 1, «8 —pe—a@.— o_o 
Furthermore the equation /, /, = 0 gives 
Che — oe oe 
And if any linear relation existed between the antecedents of the 
different /’s such as 
o=ac+684+hN4+met et... Ho 
there would result the equation 
LO OG OF Hae. i — 0; 
which contradicts the hypothesis that /, is expressed as the sum 
of the fewest possible number of dyads. And similarly in the case 
of the consequents. Hence the total number of dyads in all the 
i’s cannot exceed ~; and on the other hand, as their sum is the. 
idemfactor, it cannot.be less than z. Hence the sets 
OB, 5. Bg R. Dy SRE EN a Bey". a 
are reciprocal, that is 
(51’) TN 8 Btn eam ea 
h=)\X - ule +l +. 
Next consider the expression 
(62) @=O$J/=O(Lt+th+ht+..J=&Rt+H1+& 4+... 
The dyadics &g, dp, &,... have the property 
(52) Dg Dy = by be = by Bo =... =D, 
owing to the presence of the factors 4, J, 2,.... The dyadic 
® has now been resolved into the sum of as many dyadics as there 
are latent roots. These are all homologous with one another and 
with the original dyadic. The equations 
(52”’) G,=16la B= hh, .. 
which follow from this fact, shows that %y, %,... have the same 
antecedents and consequents as J, f),.... Hence if @ be ex- 
pressed in the form (9) where the antecedents and consequents are 
reciprocal sets, it follows that # will consist of a series of matrices 
strung along the main diagonal and equal in number to the number 
of latent roots. The further discussion of @ may be restricted to 
the treatment of these individual dyadics %g, p,.... The question 
of the reduction of a dyadic to standard form has been reduced 
to the single case in which the dyadic has only one latent root. 
Gibbs then reduced dyadics with only one latent root to a matric- 
ular form in which the terms underneath the main diagonal dis- 
appear. This reduction, however, is not complete, and consequently 
a modified form of it will be given: It may be pointed out that 
the above reduction of ® to a sum of independent dyadics is in 
no way dependent on the completeness of # The result is equally 
