552 The Ohio Naturalist. [Vol. VI, No. 8, 



3. Buds ovate, merely acute, outer pair of scales red, next pair gluti- 

 nous; twigs of the season light brown or gray, the tips withering. 



V. opuliis L. Cranberry-tree and Snowball. 



3. Buds long and rather slender; twigs of the season dark reddish brown. 



V . lentago L. Sheepberry. 

 and V. prwnifoliuni L. Black Haw. 

 V. lentago has prominently acuminate leaves and acuminate win- 

 ter buds. V. prunifolium has obtuse or merely acute 

 leaves and acute winter buds. 



4. Leaf scar narrow, V-shaped or broadly U-shaped; twigs gravisb 



brown; buds oblong. V-lantanah. Wayfaring-tree. 



4. Leaf scar broad, U-shaped to heart-shaped; twigs reddish. 



V. alnifolimn Marsh. Hobblebush 



5. Outer pair of bud scales fully half the length of the bud or more. 



V. pubcscens (Ait.) Pursh. Downy Arrow-wood, 



and V . dentatum L. Arrow-wood. 



V. pubescens has pubescent leaves with very short petioles; the 



twigs of the season are often nearly glabrous. V. dentatum 



has nearly glabrovis leaves with petioles }/i-\]A inches long. 



5. Outer pair of bud scales barely half the length of the bud or less. 



Y . molle Michx. Softleaf Arrow-wood, 

 and V . acerifolimn L. Mapleleaf Arrow-wood. 

 V. molle has pinnately veined coarsely dentate leaves. V. aceri- 

 folium has 3-ribbed, 3-lobed leaves. 



A SIMPLE FORMULA FOR MIXING ANY GRADE OF 



ALCOHOL DESIRED. 



E. W. Berger. 



Let P represent the grade per cent, of the alcohol on hand, P' 

 the grade per cent, required, v the number of volumes of water 

 to be added to one volume of P to make alcohol P' and x the num- 

 ber of volumes of P that we desire to change to P'. Then 



Px P — P' 



= P'. This gives us (1) PV=P— P', and (2) v=^ ^, ' 



x-|-vx P 



Of these, (1) gives us the pharmaceutical rule quoted by Pro- 

 fessor Schaffner in his " Labratory Outlines for General Botany " : 

 "Take of the grade at hand as many volumes as the number of 

 the per cent, you wish to make, then add to this enough volumes 

 of pure water to make the total number of volumes agree with 

 the number of the per cent, at hand. " And (2) may be translatee 

 into words as follows: — Rule: To find the number of volumes (v) 

 of water to he added to one volume of the grade per cent. (P) on hand, 

 to m,ake alcohol of the grade per cent. P', divide the difference be- 

 tween the number (P) denoting the grade per cent, on hand and the 

 nufnber (P') denoting the grade per cent, required, by the latter 



P — P' 



number (P')- Or, which is simpler, v= — ^^ — ' 



Professor Irving Hardesty in "Neurological Technique" ap- 

 proximates the above formula (2), and if he had worked it out 



