We present the results of variable star searches of the M31 dwarf spheroidal companions Andromeda I and Andromeda III using the Hubble Space Telescope .
A total of 100 variable stars were found in Andromeda I , while 56 were found in Andromeda III .
One variable found in Andromeda I and another in Andromeda III may be Population II Cepheids .
In addition to this variable in Andromeda III , another four variables are anomalous Cepheids .
So far , no definite anomalous Cepheids have been discovered in Andromeda I .
We discuss the properties of these variables with respect to those found in the other dwarf spheroidal galaxies and revisit the anomalous Cepheid period-luminosity relations .
We found 72 fundamental mode RR Lyrae stars and 26 first-overtone mode RR Lyrae stars in Andromeda I giving mean periods of 0.575 day and 0.388 day , respectively .
One likely RR Lyrae in Andromeda I remains unclassified due to a lack of F555W data .
For Andromeda III , 39 RR Lyrae stars are pulsating in the fundamental mode with a mean period of 0.657 day and 12 are in the first-overtone mode with a mean period of 0.402 day .
Using the mean metal abundances derived from the red giant branch colors , the mean RRab period for Andromeda I is consistent with the mean period - metallicity relation seen in the RR Lyrae populations of Galactic globular clusters , while Andromeda III is not , having too large a mean RRab period for its abundance .
In Andromeda I , we found two RR Lyrae stars which are noticeably fainter than the horizontal branch .
We discuss the possibility that these stars are associated with the recently discovered stellar stream in the halo of M31 .

Using various methods , we estimate the mean metallicity of the RR Lyrae stars to be \langle { [ Fe / H ] } \rangle \approx - 1.5 for Andromeda I and \approx - 1.8 for Andromeda III .
These estimates match well with other mean metallicity estimates for the galaxies .
Interestingly , a comparison of the period-amplitude diagrams for these two galaxies with other dwarf spheroidal galaxies shows that Andromeda III is lacking in shorter period , higher amplitude RR Lyrae stars .
This may be a consequence of the very red horizontal branch morphology in this dSph .
Not including the two faint RR Lyrae stars , we find \langle V _ { RR } \rangle = 25.14 \pm 0.04 mag for Andromeda I resulting in a distance of 765 \pm 25 kpc .
For Andromeda III , \langle V _ { RR } \rangle = 25.01 \pm 0.04 mag giving a distance of 740 \pm 20 kpc .
These distance estimates are consistent with those previously found for these galaxies .
We discuss the relation between the specific frequency of the anomalous Cepheids in dwarf spheroidal galaxies and the mean metallicity of the galaxy , finding that the M31 dwarf spheroidal galaxies follow the same relations as the Galactic dwarf spheroidal galaxies .
We also find that the period-luminosity relations of anomalous Cepheids and short-period Cepheids are distinct , with the short-period Cepheids having higher luminosities at a given period .