642 
Proceedings of the Royal Irish Academy. 
Solving tlie quadratic, we find the position of the two asymptotes, 
viz. : — 
,/ + 0-99608621 + &c. 
\_ 1-011960389 + &c. 
For the reasons already given 
o>'2>l, 
and as we must take points on the curve outside the asymptotes, it 
appears at first sight that we might have two real values of w', one 
positive and one negative. This, however, is not the fact, for we can 
show that the negative value is the only one that will answer. 
8. General Conclusions respecting Binary Compounds. — The two 
fundamental mechanical equations of binary compounds are (4) and 
(5), derived respectively from the Conservation of Areas, and the Loss 
or Gain of Energy during combination. 
By eliminating wi between these equations we find, in general, 
equation (6) 
)»=[(/3'-/3-l)«''+2/3o,'-/3^)±^^ = 0, (6) 
in which ± represents the loss or gain of energy (expressed in 
heat units) during combination. This equation, regarded as a curve 
of which # and oi' are the Cartesian co-ordinates has been sufficiently 
discussed, and the available value of w' shown to be negative and 
greater than unity. 
It follows from this that the available value of wi must be negative, 
and this value is found by eliminating to' from the equations (4) and 
(5), from which I find 
>»=j(^'-^-l)< + 2^a,,-ij ±^^=0. (7) 
ITow it is well-known from the discussion of the planetary pertur- 
bations that they become very serious whenever the periodic times of 
two or more planets or satellites have a small common multiple. 
following this guiding idea, and taking into consideration the 
intense affinity of hydrogen for fluorine and chlorine, I thought it 
highly probable that the ratios of the periodic times of fluorine, 
chlorine, bromine, and iodine to that of hydrogen, would turn out to 
be represented by fractions, with small integers for numerator and 
denominator, or perhaps even whole numbers. 
