153 



Reprinted from the Journal of the Franklin Institute 

 Vol. 245, No. S. May, 1948 



Printed in U. S. A. 



DIMENSIONAL ANALYSIS: AN APPROACH FROM TRANS- 

 FORMATION THEORY AND A CRITERION FOR 

 SCALING MODEL EXPERIMENTS.' 



BY 



J. C. DECnJS.2 

 1. THE INVARIANTS OF CONTINUOUS TRANSFORMATION GROUPS. 



In a recent article in this journal, Langhaar (l)' has drawn attention 

 to a method of dimensional analysis in which the kinds of physical 

 quantities are explicitly given a vector representation. The crucial 

 theorem of this subject was proved by Langhaar in terms of theorems 

 on homogeneous functions. 



The author of the present paper has formalized the subject matter 

 of dimensional analysis from a slightly different mathematical point of 

 view, in which, however, as is natural, the material receives a practically 

 identical representation. Because of the relationship of the following 

 method to a very general method of discussing problems in mathematical 

 physics, it seems worth-while to set it forth here. In addition, since 

 the de^■elopment is aimed at the treatment of scaling laws for model 

 experiments, a new criterion is given for the possibility of satisfying the 

 scaling laws when certain restrictions are imposed upon the variables. 



A fruitful approach to the problems of dimensional analysis may be 

 made by regarding any physical equation as the expression of an in- 

 variant under the transformations of some group. Frequently the 

 knowledge of some form of spatial symmetry, the requirement of in- 

 variance under permutation of indistinguishable particles, or the neces- 

 sity of special invariant properties which must obtain under transforma- 

 tions simultaneously involving temporal and spatial coordinates has 

 been used to simplify or actually to advance the mathematical descrip- 

 tion of the physical world. Although the results of the following appli- 

 cation of invariant theory are all rather well known, the method itself 

 is rather elucidating, particularly with regard to the derivation of the 

 laws of similitude which govern the scaling of model experiments. 



It will be shown below that all of the results of dimensional analysis 

 follow from the single postulate that all physical relations must be 

 expressible in a form which does not depend upon the "magnitude" of 

 the various physical units of measurement requisite for the description 

 of a given physical situation. 



• Contribution No. 393 from the Woods Hole Oceanographic Institution. 

 'Woods Hole Oceanographic Institution, Woods Hole, Mass.; now at Brown University, 

 Providence, Rhode Island. 



' The boldface numbers in parentheses refer to the list of references appended to this paper. 



