82 H.S. H. WARDLAW. 
by Neumann’s method are 4°0/% + 1°3% high. The casual 
error of the results shown here is a good deal smaller than 
those shown by the results on milk given in Tables I and 
II, a fact which seems to idicate that there may be sources 
of error in the preliminary acid-ashing process to which 
the milks were submitted. 
Sources of Error. 
In the estimation of phosphorus by the method of Neumann 
modified as described, there are, apart from the preliminary 
acid-ashing process, five stages at which sources of error 
may be sought. 
1. The precipitation. It may (a) be incomplete, (b) the 
precipitate may not have the composition assigned 
to it. 
2. The washing. (a) This may be insufficient, (b) the 
precipitate may be dissolved by the wash-water, (c) 
the precipitate may be decomposed by the wash- 
water. 
3. The boiling off of ammonia. (a) It may be incomplete, 
(b) the substances present may be altered in some way. 
4. The boiling off of carbon dioxide. (a) It may be incom- 
plete, (b) the substances present may be altered. 
5. The final titration. (a) The end point may not be 
satisfactory, (b) the value obtained will depend on 
the nature of the substances present. 
Let us now consider the possibility of error coming in at 
these several stages. 
1. (a) The completeness of the precipitation was proved 
by digesting the filtrate for several hours with excess of 
ammonium molybdate and ammonium nitrate. Mere traces 
only of further precipitation were ever obtained in this 
way, and as the precipitate contains only about 1°6% of 
phosphorus no appreciable error is introduced here. (b) At 
