592 JOHN JOHNSTON AND PAUL NIGGLI 
postulated by the phase rule. On the other hand, while in several 
cases the number of phases found does not exceed the number 
predicted from a consideration of the phase rule, on the specific 
assumptions that equilibrium has been reached and that the com- 
ponents have been correctly inferred, yet the presence of more or 
fewer phases than expected proves nothing except that the assump- 
tions were wrong. 
From the discussion of these limitations to the phase rule we 
conclude, therefore, that our present knowledge of the facts con- 
cerned is so scanty that the phase rule is not yet of much service 
in aiding us to describe the natural history of metamorphic rocks.* 
Nevertheless it may be of assistance in the classification of rocks; 
for this reason, and in order to emphasize by means of actual 
illustrations some of the statements made in the above paragraphs, 
we include a brief discussion of the application of the phase rule 
to ternary systems. 
The ternary system—lime-alumina-silica—The relative simpli- 
city of two-component systems tempts one to assume that systems of 
three or more components may somehow be considered in an equally 
simple manner. That such is not the case is amply demonstrated 
by the lime-alumina-silica series, which has recently been very 
thoroughly investigated? at constant pressure. 
From this work, in which the variables considered are only con- 
centration and temperature, it follows that there are—instead of the 
simple ternary relation with a single ternary eutectic, such as the 
textbooks commonly treat—about eighteen invariant points (at 
which 3 solid phases coexist with 1 liquid phase) any one of which . 
might under proper conditions show the usual eutectic properties 
(e.g., eutectic structure) .3 
The phase rule will of course be an indispensable adjunct when the experimental 
facts are available. 
2 E. S. Shepherd and G. A. Rankin, ‘‘ Preliminary Report on the Ternary System 
CaO—A1,0;—SiO,,” Jour. Ind. Eng. Chem., III (1911), 211; Z. anorg. Chem., LXXI 
(1911), 19. 
3 In passing it may be suggested that the so-called eutectic structure as inferred 
from the study of alloys may or may not occur in the case of silicates. It is readily 
conceivable that systems in no sense eutectic may easily yield the so-called eutectic 
structure and vice versa; cf. Shepherd and Rankin, Joc. cit. 
